Some bats do eat spiders.
Depending on the species, bats have a varied diet. Some bats eat only insects, while others feed on fruit nectar. There are other species of bats that are carnivorous eating small rodents and even other bats. And still yet there are bats that subsist solely on blood.
The majority of the bat species (between 60-70%) are insectivorous. These bats feed on insects and though spiders are not considered an insect, some insectivorous bat species do feed on them. The Northern myotis and the little brown myotis are examples of bat species that feed primarily on spiders.
“What Bats Eat | Bat Conservation.” Bat Conservation. N.p., n.d. Web. 15 Nov. 2011. <http://www.batconservation.org/drupal/art-what-bats-eat>.
Feldhamer, G., Carter, T., Whitaker, J. 2009. – Prey Consumed by Eight Species of Insectivorous Bats from Southern Illinois. The American Midland Naturalist. 162(1):43-51.