Hm, might be good to rewind a bit. You probably have long odds on Rome with 3 archers on a hill.
Your warriors are quite helpful in an early attack, and if you want to WC rush him, you really should try out a worker-steal. That will set him back a lot, and he will just keep building workers and not improving his tiles. You'll find that Rome is much easier to defeat if you'd stolen a worker. And - you can use the stolen worker to chop out a WC in Memphis sooner.
You did good with the tile improvements - so stick with that - it was really the warrior management that got you fouled up on the Rome attack.
(Great advice from elitetroops on using the lake to watch Rome also).... but parking a warrior by the horse / ivory / or wheat up north is what you want. And as he said, your fogbusting is probably overkill for this map with WC's having one warrior by the pigs is probably good enough, and possibly overkill on prince.