The place you really need to draw that circle around isn't so much the Capitol as it is the university. It's the 40,000 or so students who want to be as close to the U as possible that really impact the parking situation the most. The area around the Capitol is tough to park in, too, but the university is what you want to look at in determining how bad parking is likely to be.
This page of the city's website:
Streets & Recycling Department: City of Madison, Wisconsin provides a lot of excellent data about parking during snow emergencies. You'll want to pay special attention to the fourth section down, where the snow emergency zone is described. I'd recommend against living within that area unless you have off-street parking, or are willing and able to live without a vehicle (i.e., biking and/or busing).
Toward the east, once you get past First Street, the on-street parking pressure starts to lessen, though the area near East High will be packed from 7:30 to 3:30 during the school year. Once you're past Hwy 30, there is little or no problem with finding on-street parking.
Going west, you'll have to go a bit farther, probably almost to Midvale Blvd., to find easy on-street parking. West High on Regent Street will have the same kind of impact that East High does for a several-block radius around the school.
Do note, however, that I am referring to EASY on-street parking. You can live right on the isthmus, even very close to the university, and still park successfully on the street. It just takes time and patience, plus a willingness to move your car to the other side of the street every night from November 15 to March 15.
Good luck to you in your apartment search and move - hope you find a great spot to settle, both for you and your vehicle.