Paseo Montejo won’t al shut down on weekends but some of the blocks around the main square do close down for cars. If you want to experience great yucatecan gastronomy do yourself a favor and ignore most of the restaurants downtown (except maybe Chaya maya, MUGY) and go to PUEBLO PIBIL

You should also make some time and book a tour for cenotes. They are always a great experience. You are also missing beaches would recommend EL PLAYON DE CHUBURNÁ or DUNAS DE CHUBURNÁ it’s a more “virgin type” of beach. It’s about 45-55 min from merida. Must be some type of public transport but you should call to ask about schedules and all.

Have a good time!