Apartment hunting in Wisconsin in the winter can actually yield some unexpected deals. You will generally have less people competing for the same apartments, which can make some places accessible to you where you would usually find yourself in a competition. Because of the reduced selection however, it is a good idea to give yourself a full two months for your winter apartment hunt.
References are very important to Eau Claire landlords. Despite its population, the locals have a very strong sense of community, which means trust is very important. If you can have someone of credibility vouch for you, such as a former employer, professor or landlord, your chances of getting a good place shoot up astronomically.
Eau Claire does have a bus system, but the size of the city results in it being relatively limited in terms of both routes and hours. If you're staying in the downtown and University areas, biking and walking are completely reasonable modes of transportation. If you live further afield and not on a bus route though, a car becomes almost required.
Eau Claire has a great amusement park which is fun for people of all ages, called Action City. For a small city, it has a thriving arts scene, with three theatre companies and the Chippewa valley Symphony Orchestra. The Chippewa River State Trail provides a location for you to enjoy whatever type of outdoor recreation you enjoy, from snowmobiling to hiking. Head to the area near Barstow Street to find bars populated with a younger crowd.