Pop hood, start engine and listen carefully for hissing coming from behind the intercooler on left & right side. Hiss will indicate a vacuum leak and rotton hoses. This will easily cause what you describe. Not seen a cold start injector failure. Usually they simply plug up from old fuel varnish. Idle should be at 900-1000 RPM. Rough idle (throbbing up & down RPMs) is easily caused by vacuum leak. It's a pretty tight design & easily jacked up by rotton rubber. As engine speeds up, manifold pressure goes from negative to positive and can split old hoses. Wont see it, but you should hear 'em hissing at you when split or off-nipple.
Been there. Done that. Two cars.
How many miles you have on the chassis? aged Gaskets will turn brittle & leak oil. After 140k-150k you *must* change out your oil pump & water pump, and replace the timing belt with (preferably) a full kit - idler pulley, etc. If you see your coolant temp unsteady, replace both pumps ASAP before you lose the engine from heat.
If/when you overhaul, use a metal head gasket & forget turning it into a hot smoke-blower with turbo boost over 16 pounds. There are no major replacement parts for these cars now, and it's only gonna get worse as time marches on. Don't mistreat it, and it'll serve you dependably.
The chopped blocks available out of Japan still need strip down and complete overhaul before use. Don't let anyone tell you otherwise. The chopped blocks are a GREAT source of spare parts. I've bought 2 in the past.
I have about $14,000 in both cars just for overhaul, and a decent list of where to get the parts and proper work done, Including rebuilding the 5-speed AWD transmissions - which no one will touch 'cause they are 'not in the book'. :bangshead:
Hindsight is 20/20. Good luck! :rofl: