Narita to Haneda Travel information

Hi, I will arrive in Narita at 4:50pm Saturday and need to get to Haneda for my international departing flight at 10:15pm, that is a 5 hours 25minutes of time in between the two flights. My questions is, what is the expected time to clear the arrival process so that I can catch a bus to Haneda? and do the Limosuine buses to Haneda leave after 6pm from Narita? Any additional information would be helpful in assisting me in getting to Haneda! Thank you!

My questions is, what is the expected time to clear the arrival process so that I can catch a bus to Haneda?

Under normal circumstances about an hour from arrival until you buy the bus ticket. It can be faster, and on some busy days it can be slower.

and do the Limosuine buses to Haneda leave after 6pm from Narita?

There are lots of departures to the domestic terminals of Haneda Airport, but only very few to the international terminal:
http://www.limousinebus.co.jp/en/bus_services/narita/haneda.html

If you take a bus to the domestic terminal, you have to add another 15-20 minutes or so to get to the international terminal by train, monorail or bus.

Note also that traffic jams are not uncommon on Saturday evenings from Narita into central Tokyo. That may cause additional delays. It may be better to take the train.

In any case, this is going to somewhat risky.

Thank you for your help! If I were to change my flight day to arrive in Narita on a Sunday at the same time of 4:50pm, do you think there would be less traffic or a better chance of getting to Haneda airport?

Sunday evenings generally have higher chances of traffic jams than Saturday evenings. So, it would not be an improvement.

Okay so from what I understand, it would probably be better for me to choose a different flight all together even though i would save 500nzd by flying this way. I guess 5 hour 25mins just isn't long enough for me to make the trip to Haneda safely with enough time for me to check in for my international departure flight out of Haneda! This is sad but thank you for your help! I raelly appreciate it! :D

Under normal circumstances you should be able to make your flight. But you need just a little bit of misfortune (a delay of your incoming flight, delays at immigration or delays along the way between Narita and Haneda) and it may become a little stressful.