I think that you need to be careful in this situation. For all you know, this girl could be a 40 year old man. I suggest meeting in an extremely public area - where a lot of people are around.
I understand that you feel that you and this girl are 'meant to be'. Online relationships can lead to this, as most people feel comfortable speaking to a person that they don't truly 'know' yet. If it works out for you, then congratulations - you are a lucky guy. However, I would be wary. Sometimes online relationships are merely a facade, as many emotions and thoughts can get lost in transalation through the internet.
Make your trip to new york one that isn't just about meeting this girl. If things don't work out you don't want to feel as if you've wasted your hard earned money on something that was never meant to be. Schedule time to see the sights, go shopping and dining - as new york is a great city.
As far as money is concerned, book in advance to insure a cheap flight. Don't travel during the holidays or any other time that demand for flights may be high - this increases the price. Look into staying at a hostel as opposed to a hotel - there are many websites that can lead you to these. Also, hostels host many a foreign travelers, so who knows, maybe you'll meet a touring companion!