On Saturday, the museum parking lot off university avenue is available for free. On Friday, however, this lot is closed to non-museum guests, and you will have to use metered parking in the area. The closest affordable spaces (a quarter per hour, for up to nine hours) are on Stewart Avenue, about five minutes away from the museum. The area where parking spots are usually available is marked as ‘A’  in the google map below, with directions to the the Johnson Museum, point ‘B’:
http://maps.google.com/maps?f=d&source=s_d&saddr=Stewart+Ave&daddr=Unknown+road&hl=en&geocode=FdK6hwIdjtlw-w%3BFQK_hwIdROtw-w&mra=dme&mrcr=0&mrsp=0&sz=16&sll=42.449381,-76.49025&sspn=0.013585,0.033023&ie=UTF8&z=16

There is, of course, also the possibility to leave the car parked downtown and walk/take the bus up to campus.