Yes, your husband will be downtown. The city is trying to encourage more downtown living, but it's not really a viable community yet. On the upside, he will be right on the light rail line. So that would be one possibility, looking down in the Museum district so he could ride light rail to work. (The area just south of Downtown is still sketchy but if you get down near the museums it's nice...don't know how much rents might be.) The bus system isn't super-practical because it's so hot at least six months of the year that walking to and from the stops and waiting in the heat leaves you pretty messy.
Otherwise Midtown is close but might be expensive, Montrose and then Upper Kirby are just west of there (and there are actually some reasonable on a professional salary apartments around the fringes of River Oaks, a famous rich-people neighborhood.) The Heights is a neighborhood that has been gentrifying over the last decade or two, and Garden Oaks is a neighborhood northwest of there that is gentrifying now so a little cheaper. Rents unfortunately have been going up fast the last year or so because right now there are a lot of people who want to move inside the Loop and a lot of big apartment development projects that are still a year or so from coming online. Once that happens, the market should settle down some. I will warn you, from the highways and even the main streets, the city doesn't necessarily look great. But inside the loop if you can get onto the side streets behind the major ones, there are pretty neighborhoods with trees, nice homes, etc.
Overall, I encourage you to a) move to Houston with an open mind (it seems like you have a great attitude) and b) ask for help from locals as much as you can. The portrayal of Texas outside the state is really different from my experience of living in the state...in the big cities I have found it to be diverse ethnically and politically (Houston's mayor for the last five years is a lesbian, which surprises a lot of people) and the only time people wear cowboy gear in Houston is during Rodeo. The city, however, is not great at presenting itself for newcomers...not that people aren't friendly and welcoming, but more that it can be really hard to sort out the great neighborhoods, restaurants, events, parks etc from the visually undistinguished mass. It's not a pretty city overall (although it has pretty parts) but it is has really nice people, so ask for recommendations! The other weird thing about Houston is that there is no zoning. So you will have housing mixed right in with businesses and offices inside the Loop, unless the neighborhood has deed restrictions. That surprises some people.
Good luck!