Don't know about a hotel shuttle, but the city center is within walking distance at least outside the months of December to February. If you're going on tours, they can pick up and drop off anywhere you want.

There are also public buses which stop opposite the hotel when going downtown and more or less outside the front entrance when coming back. You can pay with exact change (get some from reception if required), an app, or tickets from certain shops.

As for KEF airport, Flybus offers tickets to all hotels for a small supplement, though you change from a big coach to smaller vans at the bus terminal (otherwise it would take forever if you had to stop at every single hotel).

Of course taxis are also available if you really want one.