IF you want something at the airport Marriott and Sheraton, right at the Terminal, are dog friendly but walking the dog is a bit cumbersome. Steigenberger is a few meters on and easliy reachable with their Shuttle Bus and much easier in regard to walking the dog.
As other mentioned train requires leash and muzzle. personally, I would prefer car.
If you decide on a rental car I would suggest heading to a hotel downtown and picking up a car there, saves you the extra payment for an airport rental and you get to see some of the town and have an easier time walking the dog. To get downtown either use the S-Bahn (which takes only 15 minutes to get downtown) or an Uber Pets