No need to go into Amsterdam Centraal, the trains to Den Haag stop at Schiphol and make one additional stop at Leiden Centraal.

There are several other threads here about buying train tickets at Schiphol that may be useful. Price for a return ticket Schiphol-Den Haag in 2nd class is EU14.40 and a one-way ticket is EU7.40. You will pay a small surcharge if you buy your ticket from an agent. No need to worry about buying in advance on this route, it is all open seating and you just get on the first train that appears.

Direct trains run about every 30 minutes at peak times and the trip takes 34 minutes. There are also intermediate trains where you need to transfer at Leiden, which is an easy transfer, but hardly worth the effort for a first-timer given the frequency of the direct trains.

If your flight is on time and things go well you should be at Den Haag well before 5:00pm. The optimistic train would be at 2:28, but a more likely option is the 2:58, which gets you into Den Haag at 3:32.