Unfortunately, there's no buses is operated after 14:40 from Haneda Airport to FujiQ.
So the only option to get there on your day arrive is, to get Shinjuku station then get the highway bus to FujiQ.
Here are the timetable.(and reservation form)
You can see only a month ahead but the timetable is still the same.
It will take about an hour from Haneda to Shinjuku by train, and approx 10 mins walk from the station exit to the bus terminal.
According to your arrival time, I recommend you to reserve at least after 18:00 not to lose your bus, and no need to rush.
Sometimes it will take time to get out from the immigration and pick up your stuff.
If you are worried to miss your bus, just book the seats and don't finish your payment online.
In that case, you will need to visit the office at bus terminal to collect the tickets and finish the payment.
So you should better to consider to take enough time to done those stuff.
( in this case, I think after 19:00 would be good idea)
Around your arrival date would be in/beginning of the rainy season in Tokyo.
If it rains hard, your flight might be delayed much more than you think.
I think not so many people will travel from Shinjuku to FujiQ by bus around that time ( night time) so that you don't have to worry to rush making your reservation.
However, please check the website above and consider if you book in advance or not and the time as your date coming closer.
Hope this can be your help!