All public transport in Moscow can be used with a Troika card including suburban trains to Novodachnaya and Dolgoprudny stations. These stations are on line D1 of Moscow Central Diameters, which is indicated on a subway map.
Estimated cost. Once you arrive in Moscow it is best to get Troika on the subway ticket terminal. In order to use it you’ll have to put some money on it. 500-1000 RUB should be enough to cover a week-long stay.
Validating. Please make sure that you validate a card on exit stations of suburban trains so that the tariffs work correctly. It only is needed on Central Diameters platforms like Dolgoprudnaya or Novodachnaya, and is not nessesary in subway or buses.
Around MIPT campus
You can find detailed information on the sights on the Tripadvisor page for Dolgoprudny:
Dolgoprudny is a rather small town that you can explore by foot or in short rides. Most of its attractions are churches, monuments and parks.
In about 3 km to the south there’s a shopping center with a sea aquarium.