Well, there is public transit from IAH, but it is rather slow....
The Route 102 bus runs on Sundays every 40 minutes and takes about 1 hr 5 mins to get to downtown Houston for US$1.25 each way.....and likely from a downtown hotel you'll find a taxi to the Museum of Fine Arts, or a connection to a bus or light rail to the Museum District.
The Route 102 schedule and map is at:
http://www.ridemetro.org/SchedulesMaps/Pdfs/102-iah.pdf
On the Sunday schedule (page 5, map on page 6), use the "time point 10" for departure time for IAH Terminal C (unsure where bus stops, or whether it stops at other terminals at IAH --call 713-635-4000 for full info)
On the schedule, "time point 10" is the downtown Houston transit center with connections to other bus routes and light rail. Further research at ridemetro.org or the museum website should tell you your options.
Route 102 bus also stops at Greenspoint Mall, which is a rather limited, empty mall, but does have a clearance oullet for the Dillards department store and a Macy's. Info at greenspointmall.com