Never too early to start looking -- starting now. June is fine too, though.
There is a significant amount of non-institute housing within walking distance of campus, or with ready bus service. Both traditional apartment buildings and converted multi-tenant houses. The Cornell Campus is very large -- figure out which section of it is "yours." As an example, a lot of independent housing is closer to the engineering quad than the official institute housing.
What exists of the Ithaca social scene is mostly clustered in Collegetown (near campus) and The Commons. Both are within walking distance of my apartment (even in January). When considering where to live, spend some time looking at a map and don't forget to pay attention to the bus routes (google TCAT). Big box stores and grocery shopping are further away, but there is an Ithaca Car Share Program (google it) that might be a good solution to your once-a-week car needs.
While the ground is solidly covered in snow three months of the year, single night snow fall is usually not that extraordinary. The roads are exceptionally well maintained, as are the sidewalks on the _major_ thoroughfares near campus.
Getting to the airport by bus is cheap and easy, but otherwise getting out of Ithaca without a car is very difficult.