How do I properly use the Bulwark?
The Bulwark is an interesting ship. It's only remarkable attributes are that if you can spend the ammo it can put out a lot of damage (shield-mitigated damage; and you start with 28 missiles) and that you start with 3 rock crewpersons. . . err crewrockpersons. Rockmen are slow as Christmas but they get 150 hp and are fire immune. They make pretty good boarders.
With 28 starting missiles and a few breaks, it's easy enough to roll through the first two sectors in the Bulwark. But if things break the other way, or if you can't find anything to suppliment your combat, you could run into trouble. The Rock has a pretty strong start I think, but it's subject to mid-game problems for the reason you mentioned; if you spend too much on ammo you'll not get other, necessary, upgrades.
In my lone Bulwark win (I've only played it a handful of times) I was lucky to find a Heavy Laser Mk II early, and then found another energy weapon in the mid sectors (I think a Blaster Mk II). Those two weapons + one launcher is a very potent combo (though that's an expensive setup at 7 power). Still, it's one you can build towards gradually as the two energy weapons can take you through most of the game.
I think it's better to save money and hope for the right upgrade than to keep buying ammo. That means there will be games where you are nearly in a fail state by sectors 3-4, but that's true even if you are buying lots of ammo. And the upside of finding good complimentary energy weapons (or getting a good drone setup) outweigh the downside.
Conserve ammo by carefully planning every shot. Use your hull missile on empty rooms (you can destroy ships with 6HP in first sector with 2 missiles instead of 3, and ships with 8HP with 2-3 shots instead of 4)
Try to save up scrap and buy teleporter early, so you can use your rock people for boarding (also getting more crew helps). This will save you a lot more missiles if you manage to kill the crew with only one-two missiles fired for weapons/medbay, then clean the rest with boarding party.
Don't fight everything you meet without thinking and consider skipping/avoiding options on certain ships like zoltans with green shield that need 3 missiles to get through + 1 missile aimed for weapons at the very least... especially if you are running low on missiles.
Buy missiles at store whenever you need to, go for options that give you more missiles when other ship surrender (that mean if they offer you fuel/drones decline it and destroy it for a chance to get missiles instead.. etc.)
Ultimately you will need a non missile weapon before the boss so try to get something on the way
The ship seems (going by experience here, and nothing more) much more likely to run into the secret sector quest chain events. In one run for instance, I had the pair of initial encounters come up four (or was it five?) times. Each time was the same. Encountered the abandoned crystal/ice ship, and then one jump away was the cookie adventure. I'm suspecting there is an intrinsic "bonus" on the ship to make the achievement easier.
Knowing this, you can somewhat build a strategy around it to a) amass free weapons or b) build an army of crystal boarders. Basically, when choosing sectors, pick those that allow for the quest encounters to spawn. That's Engi and Rock controlled sectors.