Actually it is easier than that:
Firstly from St. Paul's Road you would probably better off walking to Whiteladies Road and taking bus 1, 40, 40A, 41, 54 or 54A to the City Centre (don't get the 8 or 9 as it will take longer). Get off at the nearest bus stop to the Hippodrome theatre, then walk to bus stop "Ci" opposite the Hippodrome and catch bus 352, 353, 354, 362, 363 to Long Ashton.
DO NOT get a £4 dayrider as that is only valid within the City itself. Instead get a £6.00 First Day ticket or simply buy a day return tcket on each bus.