The metro-train to New Street is frequent and fast on the line that runs through Edgbaston. It takes about 25-30 minutes to commute from Sutton Coldfield/Four Oaks and that's quite far out from the centre.- I would look for anywhere close to any of the stations on that line.
As Raymond said 45 mins is a long way on the train so you could live in a huge radius, personally I would avoid the areas immediately adjacent to the centre, including Edgbaston, but it all depends on your budget.
I rent out Four Oaks and it aint cheap at all.
Sutton, Four Oaks, Shirley, Solihull are more for families imo and are expensive because of the good schools so you are paying rent for something I presume you won't be accessing.