If you're interested in studying Classics in Italy, have a look at the Greek & Latin Summer School run by the University of Bologna - more info here. I heard about it a few years ago, looks very interesting and I'm considering signing up for it myself - I'm in a similar position to you in terms of being half way through a gap year with very little planned!
The course you suggested actually looks really interesting. There's a big gap-year tour called Art History Abroad very popular among spoilt 'gap-yah' types and costs over £8,000 (yes, £8,000) for a two-month course. Compared with that, the British Institute is actually pretty good value so, if it's what you're passionate about doing, I'd say you should definitely go for it!