The easiest way to get to Groningen from the airport is by train. There is a connection from Schiphol Airport train station to Groningen train station twice an hour. 3 minutes past the hour there runs a direct train from Schiphol to Groningen. 33 minutes past the hour there is a train to Leeuwarden. If you take this train you can change in Zwolle to a train to Groningen, which will always be just across the platform. The trip takes two hours and ten minutes. You can purchase tickets online via the NS website. Especially for your return trip, when you have a flight to catch, I would recommend checking the NS website before you leave to make sure there are no delays.
To get from Groningen station to the University Hotel the easiest thing to do is to walk: it takes about 15 minutes and it is a straight shot. Check our map for directions. You can also take a taxi, they are in front of the train station. Be sure to have cash, they are unlikely to accept credit cards. The fare will be less than 10 euros.