contact cement should work fine, it holds the souls on your shoes, the lustrous on your car, fromica on countertops and a number of other things. regular rubber cement would be weak and a mistake, but actual contact cement should hold up well.
here is a trick to make it easier to install and make a strong bond: paint both sides to be bonded with a layer of contact cement and let dry 30 min or longer, than paint on another coat and allow to dry overnight. than put a heat lamp on both sides where there is adhesive (may take several heat lamps for a large area). Tis will "activate" the glue and soften upholstery fabric so it is will stretch into complex shapes. Than bond the two sides together, work carefully from the center outward (when it touches, that is where it will stay, so you will want to practics this step before you apply the cement). Make the fabric side larger and than use a hard roller to press and bond it to the surface, rolling it out to the edges. And than trim off the extra length.
I have also put large sheets to be glued out in the hot sun (if it is available) to activate the glue. heating the fabric usually allows you to stretch the fabric into compound curves. I have recovered classic car dash hoods and complete dash board this way without, without puckers or wrinkles, even around the instrument clusters. Never had one come off using this method. I learned this when I worked in an athletic shoe repair store when I was in high school, we used it to resoul athletic shoes, including the shoes with the rubber that wrapped up around the toe and heels of the shoe.
Also, for the fabric use quality upholstery vinyl (automotive type is generally very good) that does not have an external fabric backing, but internal, so it will not absorb moisture.
Good luck.