Years since I've been there but it was a dead loss even then and seems to got worse with some classics now ruined. So apart from getting a train to Winchester or Salisbury you should try this one
South Western Arms .
The Dolphin Inn is the other side of the railway and a main road and is meant to be good. A couple of micros are
Bookshop Alehouse and
The Butcher's Hook which is in Bitterne. All are a shortish walk or for Bitterne a longer walk or bus ride away. Down by the Town Quay
The Dancing Man looks good as it's a brewery tap but the place was the Maritime Museum when I was last drinking in Southampton and
Platform Tavern might be worth looking at too as it's nearly next door, those are probably a bus ride.