Do they use an anestheic when circumsizing a 2 month old baby?

Yes. The surgeon may opt for a general anesthetic. Some will use local anesthesia with sedation.
Some do some don't. Some physicians use topical Emla (lidocaine and prilocaine) or lmx topical anesthetic cream, applied about 20 minutes ahead of time. Some cover with tegaderm or the end of a finger of a surgical glove to keep cream in place. Some inject a little Lidocaine as an anesthetic block. We have found topical cream to be very effective plus "sweet-ease" sugar water for baby to suck on. Sucking releases endorphins and reduces pain.