There are trains from Schiphol Airport to Utrecht leaving every fifteen minutes. Travel time 30 minutes, no changeovers necessary. Further information on train connections can be found on the site of the Dutch Railways or the Journey Planner.
After midnight there are still trains, but they will take longer (50 minutes).

The train fare for the single trip from Schiphol to Utrecht CS amounts to €7.90 (2nd Class) or €15.80 (1st Class). It is not possible to buy tickets in the train (you pay extra and probably a big fine, depending on your explanation). Ticket machines should accept bank cards and credit cards. When buying the ticket at the ticket windows there is a surcharge of €0.50.

You can travel from Utrecht Central Station to KNMI or to hotel De Biltsche Hoek by taxi or by bus. KNMI is near a bus stop, but it will take a 10 minute walk from the same bus stop to get to hotel De Biltsche Hoek.
Travel by taxi is perhaps more convenient, but more costly: €15 - €25. Taxis can be found on an exterior platform at the top platform of the railway station. Taxi drivers should know the hotel.

Royal Netherlands Meteorological Institute/KNMI
Wilhelminalaan 10
3732 GK De Bilt
Tel +31302206911
http://www.knmi.nl/

Hotel De Biltsche Hoek
De Holle Bilt 1
3732 HM De Bilt
Tel +31302205811
http://www.biltschehoek.com/home.aspx

There are buses leaving Utrecht Central Station for De Bilt roughly every 10 - 20 minutes. The numbers are: 50, 52, 53, 74. These are regional buses (called ‘streekvervoer’) and they start in front of the station. Walk from the top platform of the Central Station in the direction of the town centre, turn right to go downstairs to the bus station, which you will see through the windows. Follow the indicators ‘streekbusstation’.
The buses have various regional destinations, so do not get confused by that. The bus numbers are indicated on the buses. Bus times are accessible through the 9292 Journey Planner.

You can buy a ticket at the bus.

The bus stop for KNMI and Hotel De Biltse Hoek is called ‘Tunneltje De Bilt’, which refers to a small bicycle tunnel at the stop, see the map. You could try your Dutch and mention KNMI (pronounced as Kah N M Ee). After a few bus stops in town, you leave the city of Utrecht to the east direction and then it is still a few kilometres on a straight road. Bus travel time about 20 minutes.

KNMI is located at 300 meters from the bus stop (see the map). You may be able to see the tower with the white radar dome on top.

To go to the hotel walk 30 meters away from the main road towards a little farm, then walk back into the small bicycle tunnel at your left hand side and cross the road. After about 150 meters turn right into a main street, cross the traffic lights and walk another 500 meters towards the hotel.

Travelling back to Utrecht Central Station can also be done by taxi or by bus.
If you prefer a taxi, order one at the hotel or at the KNMI-reception.

Going back by bus is straightforward. The bus stops at the north side of the tunnel, so coming from KNMI you should walk through the tunnel. There is also a bus stop at the Hotel De Biltse Hoek from where you can go directly to the Utrecht Central Station (only in this direction).

See the map below, you can also use http://route.anwb.nl/routeplanner/.