That's the Dean Tavern. Easy to get to and back from by public transport - any Gorebridge or Birkenside bus from the city centre, there's a stop across the road from the Dean. Last bus back is after the session finishes. You get to see some countryside on the way.

I live in Newtongrange, five minutes' walk away, but haven't been to that for a few years - it's a very large singaround with a lot of singer-songwriter stuff, mostly nice people but not really my thing. Usually the best variety of the regular Midlothian session/singarounds, with a reasonable amount of Scottish material (the others are overwhelmingly American singer/songwriter). Gillian is a good singer.