Answer
Yes, it is possible to live right on the Rambla in Montevideo: there are waterfront homes, and people do choose to live there. It is not just a promenade but also a residential stretch with apartments.
This location has its own considerations. The Rambla offers access to the coast and open space, but also brings wind, salt air, activity along the waterfront, and sometimes higher costs. For a waterfront apartment, the windows, heating, and actual noise levels are particularly important.
Before choosing an address, walk the route yourself and check the building entrance, the crossing over the wide road, nearby shops and transit stops, and your route home in the evening. If the sea really is part of daily life, the Rambla may be convenient. If not, a nearby neighbourhood can sometimes offer more space for the same budget.