There are many hotels 5 minutes from the airport in a development called Aerocity. The Four points is good but a bit farther and for a very late arrival, closer is better in my opinion.

I would suggest Red Fox which is cheap but comfortable western style - but there are others. They offer a package with r/t transfer. Typically the driver would be waiting for you after you exit arrivals (inside the arrivals hall) but they don't always show up and this requires a call or for you to take your own cab. Many drivers for the luxury hotels in the arrivals hall will be willing to do this for you - as they have for me several times when my phone didn't work and the driver didn't show.

To book a car if necessary, use one of the transport desks inside the arrivals hall (to your left). You will typically get an older car but they tend not to have an issue with such a short ride - the taxi companies outside (e.g. Meru) often do.

No worries, the process is easy and safe.