It's a good 30 minute walk into town and usually very hot and humid; it can be done, but I think you would be better off taking a cab (about $10--less if you share) and saving your energy for when you get there. The beach downtown is public and free, but not very good for swimming--rocky and not too clean; but fine for just kicking off your zapatos and strolling.
The good swimming beaches are south of the river and mostly private; but the resorts will usually let cruise passengers use them--there may be a small charge to use the facilities.
The shops and cantinas along the Malecon are lots of fun; and the Cathedral and City Hall are in the Plaza in the center of town . . .just a couple of blocks from the beach. And the artists' colony that has built up around the river is a must.