The short answer
Yes, you can enter Hotel Argentino without booking a room to view the public spaces. This makes the hotel an architectural landmark in Piriapolis, not just a place for guests to stay, and it can be included in a short walk around the city.
Inside, it is worth paying attention to the lobby, which is decorated with stained glass. The décor also includes elements of alchemical symbolism, making a visit a small stop in its own right on a route around the city.
The hotel remains operational. Even if entry is allowed, you should behave as you would in a partially public space: do not disturb guests, do not enter restricted areas, and treat the visit as a brief introduction to the historic interior.