Haven't done it with Toyotas - but Hondas are popular vehicles to perform engine / transmissions swaps. It is possible, because the Corolla was manufactured and sold with either transmissions, generally the bolt patterns would matchup with the block and not much modification is needed to be done to the chAssis. Cost will depend on how you source your part - get a refurbished unit with warranty, look to spend anywhere from $650 and up - from a salvage yard, depending on its condition - looking at $250 and up. Doing the swap is more time consuming than anything else - engine out of the car is easier, but if you have a decent floor jack (to hold up the tranny) and the car on some ramps, you can do it with the engine in place. But be prepared for possible suprises and other parts. Example of what replacement parts that might be involved in the engine swap:
Manual transmission, Flywheel, Pressure plate, Clutch disc, Throwout bearing,
Clutch cable, Clutch pedal assembly, Brake pedal assembly, Engine/Transmission crossmember, Transmission harness (Engine harness 2), Shifter boot and Associated trim, Shifter assembly, Shifter trim bezel, Rear (Firewall) engine mount
Rear (Firewall) engine mount bracket, Transmission mount or bracket, Starter,
Clutch interlock switch, Neutral safety switch, Speedometer pinion gear and Associated guages or sending units, ECM upgrade or reprogramming, Rear main seal, Axle seals, Input shaft seal, possibly more.
Might be easier to due a complete engine/tranny swap together and then drop that into your chAssis. Might set you back about $2000 to start, but less problems in the long run. Keep in mind that a manual will give you "crisper" performance, it may not be enough when paired with the 1.6L engine.
If you just wanted to get a standard transmission - an alternative to doing all of this or paying someone to do this for you - you could pick up a 1993 Corolla w/standard transmission with some cosmetic damage (ripped interior, collision damage, etc.) and strip your exisiting car for the parts - sell what was left to a salvage yard and end up paying less than what you could have for just the transmission. Just an option.
Hope this help out.