The Metro-North goes from White Plains to Grand Central Station in 35 minutes.
By the time you check into the hotel, it will probably be around 10 PM. There are trains that run that late, but it doesn't really seem like you have the time to do anything especially if you have to be up by 6 AM.
Here is the Metro-North website. Put in White Plains, the date of your trip, and the time to see what trains are available. You can also get a reverse schedule.
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