Well the thing is that it will cost you one way or the other. It will only be sensible if you'd do atleast most of it by yourself first of all. Also you never know in what state the engine you intend to put in is. This would make sense if you'd intend to first fully rebuild the engine, but that is a pita and costly job and won't be sensible for most people. Then you will get also some unusual faults that will be very hard to diagnose due to it being a transplant. So it financially doesn't make much sense. It will in the end become cheaper to buy a 1.8T A3/S3 and transfer all the special bits and pieces (if you have any) from your N/A A3 to the turbo one and sell what else is left over of the N/A.
However if you're hands on with cars and think the whole process would interest you or want to get some real experience then I guess go for it. If you have a DBW as in a so called electronic pedal, then you either have to find a 1.8T engine from facelift cars that have DBW as well (still could be connection issues that I don't know, however in the big picture it should work). As far as I know only AGU is DBC from the A3 engines (not counting in pre-fl S3 engines), so anything besides AGU could work (unless you don't want to change the pedal etc. which would be probably somewhat pita).
EDIT: As I have a 1.8N/A myself, then trust me I've considered this countless times myself and it never in the end has made sense for me. Too much hassle and not enough gain. If you're current engine is in a bad state then consider that the engine you get could be in a similar state or worse (or better). It's a lottery really.
