The decline in port-related activity in Punta Arenas has led the city to convert and redirect its economy towards a much more lucrative sector: tourism. The city is well known amongst tour operators, especially as a strategic port of call for cruise ships.
This "sandy point" (etymologically) is, however, not really up to the standard of the ten or so other possible stopovers situated along the Pacific coast. Its beaches are less attractive and, especially, polluted; and the climate is stuffy, with the breeze restricted by the Nicoya Peninsula, which lies facing it. As for the city center, there's really nothing to say about it.
In reality, the main thing the city has to offer its close proximity to San José, which has made it the main place to go for people from the capital hoping to do some sunbathing at the weekend.
So, unless you want to include an excursion to a large center for all-inclusive hotel tourism in your trip to Costa Rica, give it a miss. Head to the port and catch the ferry to the southern part of the Nicoya Peninsula.