It's going to be difficult to find much to do on Jan 1 in Houston.... the Menil art museum is closed...and there is a major football game in Houston starting at noon.
Yes, renting a car will likely be necessary to go anywhere that's open on Jan. 1....at IAH there is a central car rental building for most car rental companies....it's about a 10 in bus ride to the central car rental building
For Jan 1 with most sites closed, I'm trying to think of historic areas which are worth seeing/walking around even if the stores/buildings are closed..... maybe look at Galveston on the Gulf coast (about 90 mins south of IAH), which has historic buildings, beach and park areas
.....About 45 mins north of IAH is a planned community called The Woodlands, which should have some nice areas for walking around...see this website for "events" calendar
http://www.woodlandsonline.com/
Also, north of IAH is the town of Spring, TX (about 30 mins from IAH) which has a "old town" historic area
http://www.oldtownspring.com/
Hope this is helpful!