Despite my initial skepticism on this matter (see the comments above), it does turn out to be possible to take public transit from the Tampa airport to Sarasota, via six (!) separate buses:
Hillsborough Area Rapid Transit (HART), Route 30, from TPA to Marion Transit Center in downtown Tampa.
Pinellas Suncoast Transit Authority (PSTA), Route 100x, from Marion Transit Center to Gateway Mall in St. Petersburg.
A combination of PSTA buses (at least two) from Gateway Mall to Grand Central Station in St. Petersburg.
Manatee County Transit, Route 203x ("Skyway Connexion") from Grand Central Station to DeSoto Station in Brandenton.
Manatee County Transit, Route 99 from DeSoto Station to downtown Sarasota.
Note that the Skyway Connexion bus only runs twice a day on weekdays (10:58 AM & 3:58 PM from Grand Central); if, and does not run at all on weekends. If those times doesn't work with your incoming flight time, you're probably out of luck.
This whole thing would be a lot easier if there was a direct connection between the Manatee County (Bradenton) & Hillsborough County (Tampa) bus systems; but as far as I can tell, no such connection exists.