Which heroes are you using? Because honestly, you will never win against the majority of the encounters by yourself with some of the hero choices.
I've been playing for a couple of months and have tried a lot of different combos with the game. Leave your tactics cards in the box - that is the "support" sphere and cannot EVER win when used in solo games.
Likewise for almost all of the Leadership cards. You can splash Theodred or Gloin in the hero pile if you want but to do so means you will have to play a 3-sphere combo and without Song cards it is almost impossible.
A solo deck, regrettably, MUST be centered around Lore/Spirit, if you want to have a chance at winning. And you pretty much have to use the "tier 1" heroes. This means:
Beravor is awesome. Theodred is awesome. The Lore guy that lets you look at the top card of the encounter deck is awesome. But I have tried them all, and the fastest, most effective route to winning overall (I am talking about all of the quests) by yourself is with the combo above. Your threat starts at 24, which is low enough to allow Dunhere to fight the enemies in the staging area. Eowyn is your quester with Bifur as back-up when needed. But Bifur is your blocker for the times an enemy comes your way. In terms of resources, the three of them interact very well. Bifur can get 2 resources a round, and the double spirit heroes provide 2 per round as well. I've had turns where I didn't do anything except move tokens to Bifur, and I never have problems playing any of the cards in my hand.
In solo mode, there are only 2 things you need to rely on - threat reduction and healing. That's it. Ironically, the spheres that specialize in those abilities the strongest also happen to have the best questing allies. And most of those questers can fight, too, which is nice. I don't know what your deck build is, but let me walk you through mine. I have been tweaking it from the time The Redhorn Gate came out, and I am actually at a point where I am getting bored with the game because I hardly ever lose.
I don't know the name of every card but I can hit the highlights of what's in my deck.
Gandalf x3, obviously. Every player should have 3 of him. He is also the only 5-cost card in the deck, or at least the only one with 3 copies. There is one copy of the guy who lets you look at the top 3 cards. I've been able to play him every game, even just having Bifur.
West Road Traveller x3 - another must have. Use these to swap disadvantageous locales with ones that provide a boon. For example, I will travel to Old Forest Road and then swap it for the one that makes me discard 2 cards at random as part of the travel cost, thus bypassing the penalty.
Heimdall - I think that's his name. He costs 1 lore and lets you look at the top card of the encounter deck. I was crazy enough to buy 2 starters, so I run both copies of him. He is a life-saver. At the end of every turn, he looks at the top card so I know how many cards to commit to the quest on the next turn. Sometimes, he blocks and dies when something unanticipated happens. But that's okay, because I have...
Stand and Fight x3 - I almost took this card out of the deck when I learned you can't use it to bring Gandalf back, but I am glad I kept it in there. It can pull back cheap allies who can block a second time or, as in the case of Heimdall, go back to performing its standard function for the deck.
The 2-cost Lore card drawer - Again, I run 2 copies of this guy. When I draw the second one, it gets discarded to Eowyn for the quest bonus, or to...
Protector of Lorien x2 - Bonus to defense, bonus to quest. I've actually won games because of this card. I would dump my whole hand into the discard pile to get the last tokens on the quest, or to keep Bifur alive.
The event healing card x3 - I used to use the Daughters of wherever, but this card is better. You can heal any character, not just heroes. In team games, I will toss the Daughters back in, but they aren't really necessary for 1-player games. I realized a while ago I am a better player without them as a crutch.
Greetings from the Galadrhin x3 - I tried this with 2 copies and it worked okay, but you wouldn't believe what that 3rd copy does for the game.
Test of Will x3 - If this isn't in your opening hand, reshuffle.
Hasty Strike x3 - I was on the fence about these since I also run 3 copies of A Burning Brand, but I hate Shadow effects, and Bifur can't block everything. This has saved my butt many, many times.
Dwarven Tomb x2 - If I had a 3rd copy, I'd run it. Pulls back the greetings. Pulls back Spirit allies. Pulls back Test of Will or Hasty Stroke. Fantastic!
Ally protectors something or other x3 - These are the guys who quest for 4 but then die at the end of the round. When Watcher in the Water comes out, I'm dropping them for the Arwen card and whatever other goodies are in the set. But right now they play nicely with Stand in Fight when I really, really need a quester.
Mirkwood Runner x3 - 3 Defense? No problem, you take 2 damage! 7 Defense? No problem, you take 2 damage!! These guys cut through anything when they fight alone, and with the help of Dunhere and Gandalf they frequently deal the killing blow to a bothersome Hill Troll or some other enemy that's been sitting in the staging area for a few rounds waiting to be pulled.
A Burning Brand x 3 - Bifur blocks. Bifur cancels your shadow. Additional copies provide questing power bonuses.
Bofur - I like to play him for 1 spirit on a round when I know I will quest unsuccessfully. His stats rock. If he's not questing, he's blocking or attacking something. The second copy is my replacement for when he dies the first time.
Miner of the Iron Hills x2 - This is the only card I drop sometimes, if I know the quest I'm facing has no conditions. But he saves the day against ones that do run conditions when you don't have Test of Will in your hand or want to save it for something else.
Dwarven Craftsman x3 - I discard my attachment for +1 Willpower. I play him and put it back in my hand. He blocks. He dies. I discard an attachment for +1 Willpower. I play Stand and Fight and put him back in play. I retrieve my attachment. He blocks. He dies. I discard my attachment for +1 Willpower. You get the idea.
Northern Tracker x2 - This guy needs no explanation. You can get overwhelmed by locations very fast and he takes care of it.
I'm sure there are other cards, but that is the basis for the deck. I expect to make changes this week (I have proxied Arwen and she is amazing!), but I won't know all of them until I see the card list.
I'd like to focus now on the Passage to Mirkwood quest that you are having trouble with. Let me tell you about a typical round 1 for me. I draw my hand, and play at least 1 ally, preferably a 2-cost Lore one like Gleowine, who can draw me a card or block in a pinch. This lets me keep a spirit resource handy for Test of Will or Hasty Stroke. Even better, it sets me up for round 2 if I don't need to play one of those cards (or don't have them in my hand). I can use the one saved spirit token with the 2 new ones I get and play Greeting to reduce my threat. Really, it doesn't matter what ally it is. The only time I might not is play an ally if I have Gandalf in my opening hand, but truthfully I have gotten to a point with it that there are some games I just discard him. He's awesome but he's not always necessary. The biggest mistake I made early on was to play him as soon as I drew him AND had 5 resources. I don't do that anymore. If I don't need to lower my threat or for him to kill an enemy, then it stays put in my hand. I almost never use him to draw cards, either. Frankly, he could have that option left off and cost 3 instead. But anyway, back to the subject.
So I play an ally, hopefully something that can quest. If I have Bofur, then I spend one spirit as a quest action and drop him into the quest. The opening threat is three, so questing with Eowyn, the ally (usually a Willpower of 1-2), and Bifur, I'm hitting it for 7-8. Flip the card, deal with the effects and then Dunhere hits the spider. The worst that can happen is I flip the "boss" spider with the threat of 32 and he saps my willpower. But that's what Test of Will is for. Or I just discard to Eoywn to offset it. That happens so rarely, though. Outside of that worst-case scenario, the most common horror is flipping the location with the threat of 3 that requires you to discard 2 cards at random. I used to go ahead and do it but not anymore. I just play around it until the West Road Traveller or the Northern Tracker can take it out.
So we've quested. We added 1-2 progress tokens, typically. Then, it's time to travel. If I flipped a low-threat enemy that I know I will have to block, it's off to the Old Forest Road we go and Bifur readies. If not, then Dunhere just smacks the Spider for 2 damage and its onto round 2. This is where you either have to throw down a second ally or reduce your threat. The deck almost always plays itself, so the decision should be fairly obvious. Quest with Eowyn and the ally with the highest willpower (or the one that can't do anything special like draw cards or offer foresight about the encounter deck). Use Bifur to block the spider, and Dunhere to attack it for 1 more damage. You may have to do this a 3rd time if you don't have another attacker. If you have the Mirkwood Runner, send Dunhere against whatever is in the staging area and let the Runner deal with the spider.
The only time I lose against this quest is when the Forest Spider is engaged and I flip the King Spider without a Test in my hand, but that might be once every 20 games or so.
Finally, don't let the "challenge ratings" get you down. I disagree with the scores most of them are given, and I have wondered sometimes who came up with the number. Hunt for Gollum is a 4, but I found it easier than Passage to Mirkwood. The same is true for Conflict at the Carrock. Of course, you have to play some variant of THIS deck to have that level of success, at least that has been my experience.
Anyway, I hope that helps. Don't give up, you will get there. Happy hunting!