Proclamation is very close and Grey Sail is close. Both worth checking out. Mews Tavern is very close, has a ton of taps, some good local options, but most are run of the mill craft-ish. Food is pretty good, does get crowded.
Whalers in Wakefield is also decent. Try Wickford Package store - they usually have a decent selection of local and regional stuff. My buddy Chris works the taproom at Revival (about 25 min north of Gansett) and is always spinning some sick classic vinyl - he posts every record on Insta @steeley_chris Over in Newport is Newport Storm. They occasionally sponsor an Improv Theatre company called "The Bit Players". Sometimes means free beer if you go to a Friday or Saturday night show (which are hilarious shows, btw). Regardless, the Bits are BYOB, so find some good stuff and bring it with you and have a laugh.
Strong recommendation for Proclamation Ale. Pretty close by is Tilted Barn, also worth a visit. Proclamation just had their 1st can release Wednesday, I would check their FB to see if cans last into Saturday. Grey Sail in Westerly is about a half hour drive from Narragansett. If you go you will be pretty close to Beer'd Brewing just over the state line in Stonington, CT.