Well can you answer the rest of this ?
Where do you arrive at '3pm' ? At the airport or at your accomodation ? Where is the accomodation located ? Is it on Manhattan next to a subway?... The subway runs all night. How many hours do you want to stay out after arriving ?...I'm assuming you may have to return to the airport for a connecting flight ? Basically how many hours can you allocate to seeing and experiencing Manhattan ?
Would you be looking for somewhere to eat , shopping or to do some fast sightseeing ?
If you want to see as much as possible of Manhattan then dont even bother to check in at your hotel. Shower and shave before your flight! If arriving in JFK then put your luggage in storage, head for the 'airtrain' [broken link removed] which is what they call the monorail which connects the airport with the subway. It costs 5 dollars to enter. Take the airtrain to Howard beach where it connects with the subway. From there you can see the whole of Manhattan! The subway costs only 2 dollars to enter. Once inside you can go wherever you want and its free to exit.
Pick up some subway maps along the way and study them well.
http://www.mta.info/mta has maps especially of the Manhattan subway and you should know in advance where you are arriving , staying and departing from.
It's important to do as much as possible before daylight fades and new opportunities replace old ones.
Take the Subway downtown to the end of Manhattan to South Ferry to see the Statue of Liberty. To be in NYC and not see the Statue of Liberty is unpardonable !
Around south ferry there is a cruise you may take for a 1 or 2 hour tour of the coast complete with a guide who points out what everything is and its significance. There is also a speedboat tour which does it all in 20 or 30 mins to the sound of rock music being pumped out.
You may also get a boat to Liberty Island where the Statue of Liberty is but the queues and crowds there are pretty time consuming so you probably wouldnt have time.
From there you can hop back on the subway and get off at canal Street nearby
http://www.nychinatown.org/canal2.html and explore Chinatown.
Tip when buying anything simply walk away and look as disinterested as possible as they love to haggle. I've managed to get stuff for 15% of what they originally asked for. From there you can go to 12th Avenue and Greenwich village.
Theres just too much to write!. Depending on the date there may be special festivals,events (jazz or otherwise) happening on the night you arrive. Do some research You have 3 weeks to do it . You should also Google for a forum populated by New Yorkers and ask them their views.