Fallen Empires tend to be paper tigers, even when they're reawakened. As long as you kit out your ships to a build that counters their inherent advantages, your fleets should be able to punch well above their weight.
My advice:
Fallen Empires tend to utilise ships with long range, loads of fighters, and lots of shields and armour.
- You can neutralise enemy fighters by using your own fighters and flak batteries to shred them.
You can negate shields and armour by using Plasma weaponry, which ties in to the next bit.
You should try to engage the enemy at point blank range to negate their range advantage.
The first two points you can manage by juggling around the modules you install on your ships. The last one can be a bit harder to pull off. If you can't negate their range advantage, the best bet is to kit out your battleships with Giga-Cannons, since they tend to outrange Plasma lances, and hope to high hell that you can last long enough to start pummeling them with your plasma cannons.
Strategically, fortifying home and pushing them into a disadvantageous situation is a good move, but ideally you should be fighting offensively. Send in a deep strike fleet and trash his space ports, never lingering long enough to be intercepted. This will have a twofold advantage for you. For one, it forces the enemy to divert fleets to intercept your deep strikers, and it'll reduce his ability to replace losses while pushing him into a defensive posture. The AI tends to fortify up in the middle of a system, which means you're almost always never far from wrecking any nearby spaceports and then immediately legging it for the next target.
Once you've wrecked his production capacity, any losses he takes are crippling. Keep smashing anything he rebuilds.
Beyond that, hopefully you have a decent stockpile of minerals from which you can recover your losses and continue to grind him down. Utilise multiple worlds for parallel production and rally them into a giant doomfleet and try to aim for a battle where you can crush his forces decisively. Occupy outlying worlds in a hit and run fashion as best you can, try not to worry about holding ground; the aim is to take ground faster than the AI can retake it and force him into capitulation.
All up, you need to have at least two fleets, preferably more. The primary fleet is your main concentration of forces. For Fallen or Awakened Empires, around 100k+ Fleet power is needed, preferably with an experienced admiral. Don't worry too much if that number drops when you kit them out with flak weapons. Their damage is low, but completely neutralising enemy fighters means your ships will last longer to punch back. The primary fleet is intended to take the enemy's doomfleets head on and, if not win, maul him badly enough to wish he'd never set foot in that system.
The secondary fleet has the job of deep strikes and harassment as outlined above. Remove his ability to produce replacements, capture distant settlements to force his attention elsewhere, and generally be such a nuisance that he has to draw off a disproportionate force to stop it from wreaking havoc on his home turf. Recommend around 20 - 50k fleet power, though it can be as low as 5 - 10k if it accomplishes the task at hand.
Beyond that, know where the enemy will come and go. Most fallen empires have Psi-Jump Drives, but will generally hit your nearest colonies to their territories. Arrange fortress flowers in your vulnerable colonies, at least three of them, and send your fleet down on his head when he pokes his doomfleet into your territory. The more densely your fortify the system, the more time his fleet will waste demolishing your forts over wrecking your fleets when in combat. Remember to customise your forts for maximum durability; their role is to slow his ass down and they are eminently expendable at the end of the day.
If you have to go over fleet limits in order to defeat him, then so be it. You'll lose those ships soon enough, and it's preferable to lose out on maintenance penalties and wreck his fleet than to have a good economy with no ships to protect it when he wrecks you, then takes your clay because you couldn't hold him off.
Do not rely on the AI for anything other than being incompetent. Awakened Empires gain significant bonuses (Cheats) to help them wreck both you and them. If you have the awakened empire on the backfoot, then make sure to keep them there by declaring war early and often. Take from them everything they have and crush them into the dustbin of history where they belong. The earlier you do this, the better, provided you have the fleet and technology to do so.
If you don't have battleships or the technology I've mentioned at this point in the game, GG.