Hi Claire,
Long term rentals are not a problem. Some landlords prefer to keep the tenancies empty for most of the year and rent them out for a couple of months in the vacation season, others rent year-round. Either way, the $$ work out to be pretty much the same for them. It's short-term rentals that are relatively difficult to obtain.
A basic monthly budget for two is ($US):
Food: 500,
Phone/Internet: 40,
Cable TV: 30,
Electricity: 150,
Water: 20.
Then there's all the other stuff you may or may not need: Car (gas), health insurance or medical services, personal services and preferences...
Renting can be tricky and there are some gotchas. If you wish assistance in selection for Punta or Piriapolis I am happy to help you as that is what I do. Send me a PM if interested.
Some personal observations about Punta vs. Piriapolis... I find, contrary to what one might think, things to be generally cheaper and easier to find in Punta del Este - groceries and daily needs things e.g. As far as living spaces, there is better selection in Punta. In terms of living, Piriapolis has a small-town feel, is visually appealing geographically and in some ways architecturally and has an interesting history. It also has a vibrant and growing expat community with a hippy influence. Punta has more of a modern feel, is a good base for exploring but is more spread out. The transit system is quite good if you don't wish to buy a car.
Hope this helps