Well hey, you'll be spoilt for choice. I think the nearest to you is The Hebrides Bar on Market Street, which can be a little gem but can be a bit rough too... worth a look.
Sandy Bell's on Forrest Road (go up the Mound, along George IV Bridge and turn right when you get to the fork in the road) is pretty much guaranteed to have a good session any night you go in.
The Ensign Ewart at the top of the Royal Mile/Lawnmarket, just below the Castle, is good on a Saturday night.
The Antiquary bar on St Steven's Street is a five minute taxi ride away and if you can get there tonight (Thursday) you'll find a good session there too. The rest are within easy walking distance of you.

Sessions in Edinburgh generally start at 9pm. Have fun!