Having just returned from Nessebar (Just down the road from Sunny Beach), I can confirm that there are loads of bars advertising live sport.

I saw the Rugby at The Funny Pub and also visited the Red Lion to catch something, although it didn't work out as I had messed up my timings.

I visited them simply because I spotted them. If you are spending a bit of time in SB I am sure you will find plenty of others.

Hope this helps.

T.