First off, I would rent a car. While nothing is too spread out, you will need to get to and from the airport by cab, which could run you close to $30-40 one way. Renting a compact car will be just about the same cost and give you flexibility to get around. It is about a 20-30 minute drive from Sarasota airport to Siesta Key beaches. It is over an hour from Tampa.
If you want to be able to walk to things, then you would probably like to rent a condo on the beach near the north end of the island/near the village. There are several Siesta Vacation rental sites that you can google and search.
As long as you are not advertising the fact that you are "alone", you should be plenty safe hanging out at the beach and getting around at night by yourself.
Look on VRBO and see if there are any studio gulf front apartments. You should be able to find something for $500 for the week. Some places may not rent for less than a week in April - it is the busy season.
Have a great trip!