You don't say which Mk2 you're planning to put it in.
If you already have a 1.8 16v, then I'd weigh up if it's really worth changing the engine as apposed to doing some quality engine mods to get more power out of the original engine.
Factory spec has the 1.8 16v at 138bhp and the 2.0 16v at 150bhp. So a few mods could get the 1.8 up to the same power as the 2.0 engine.
The mk3 2.0 16v engine is a complex (well to me anyway) beast with an ecu controlling it all which makes it slightly more prone to going wrong.
And from my experience owning a Mk3 16v and a Mk2 16v, I prefer the Mk2's engine. It's simpler, more raw, less to go wrong. Mind you I had quite a troublesome Mk3 (breakdowns, stalling, new custom cat conv, new dizzy, new clutch.....) and not all Mk3s will be like that.
If you're starting with a non-GTI mk2, you'll probably also need to get new engine mounts, new brakes and such.
A mk3 engine will need it's own wiring loom, I don't think you can use a mk2 loom with it, and I'm not sure whether you can use the same pedal box either.
You'd also have to think about the gearbox and clutch. If you have an old and worn mk2 gearbox and clutch you probably don't want to run a newer mk3 engine through it.
If you can get a 2.0 16v engine with gearbox and clutch for ?500 or less that's good, but you'll then need a couple of days of mechanic's time to do the conversion, assuming no problems. (My usual garage recently charged a mate of mine ?800 for a straight like-for-like engine swap when his engine blew up and that's including the engine for a V8 landrover)
My thoughts are, if you can get a good engine cheap and get the job done easily/cheaply but done well, then go for it. But if you already have a decent 1.8 16v then I'd spend the money modding/tuning that instead.