There is a train station at the airport in Amsterdam Schiphol Airport.

From there take the first train to Amsterdam Centraal and from there there are every 2 hours ICE high speed trains to Cologne and Frankfurt/Main or IC long distance trains via Hannover to Berlin. Also nighttrains start in Amsterdam to Munich but also to Switzerland (Basel, Zurich) or Austria (Innsbruck, Vienna). 

A slightly faster route is to take an IC train to UTRecht (depending on what destination) and change there=avoid the chaotic and overcrowded Ams. OR to Amersfoort if going on th Berlin train.

The station is actually underneath the airport and has 3 platforms with 6 tracks with so many trains that many unaccustomed people get annoyed or confused.

Thanks for the info.  Our final destination will be Budapest, Hungary.  Do you have specific over nite trains leaving Amsterdam that go through Germany and Austria and end up in Budapest?    

The furthest an overnight train from Amsterdam goes is Vienna.

Berlin or Munich are the westernmost starts for an overnight train to Budapest.

Thank You.  When is the best time to buy a eurail pass and a reservation?  Days before travel or the days of travel?  

How far in advance you need to make the reservations depends on when you plan to travel. If you want to travel during the summer you should make the reservations ASAP.

If you will travel during low season there will be less demand on the trains.

When do you plan to travel?

You can buy a pass minutes before taking your first train. For reservations, it depends a lot on the country and exact train.

Our daughter works for Delta Airlines so we can get free stand by travel.  We plan on leaving the USA and arriving in Amsterdam early the week of July 25, 2022.  Once in Amsterdam, we want to buy a Eurail Pass and board a night train as far as we can go on our way to our final destination in Budapest, Hungary.  We assume we can take other trains easily to get to Budapest.  We need to be at the  Budapest airport iBUD] early on Friday, July 29, 2022.  

Ok. Then you should make your reservations ASAP as that period is during the high season for travels. There is a big risk that the night train is sold out already. 

You don't need to buy the pass before making the reservations so you can make the reservations first so that you know that you can travel as planned. 

If you give as much details (departure date, time and route) you can about how you want to travel, then you will get advice here about where to make the reservations. 

If you are only taking only one journey from Amsterdam to Budapest, than you book only a normal train ticket and not the Eurail Pass.

For Tickets you can check german railways bahn.com wich should book you all day trains on one ticket or the new online shop of Hungarian railways MÁV-csoport | (mavcsoport.hu) wich has really cheap tickets for the same trains also from the Netherlands. 

For Nightrains you can check routes and prices on nightjet.com For nightrains, book asap because they are weeks full due to high demand.

Here some examples for day trains via bahn.com,

The international ticket office at Schiphol airport should be able to sell Eurail passes. However, booking night trains is an entirely different matter this summer (tested for 2 persons):

Amsterdam - Vienna: fully booked for departures 25-28 July.

Amsterdam - Munich - Innsbruck: only seats available for departures 25-27 July, fully booked on 28 July.

Amsterdam - Zurich: only seats available for departures 25-28 July.

Amsterdam - Basel: 25+28 July only seats, 26+27 July couchette or seats.

When booking on the day of departure I'd expect only seats to be available, if anything.

Additionally, if this is the only train journey you'll be making, buying a normal ticket instead of a pass makes more sense.