If she was planning on taking Amtrak to CT (which runs to New London on the shoreline) she would have had to take Amtrak from Penn Station.
Again, I highly doubt Amtrak was running to where she was going at that hour either. Metro North runs about every hour until around 1 :00 A.M.
It confuses me as to why she was at Port Authority at ALL, if she was planning on taking a "train" which is what she said to Missy. She never mentioned a bus that I know of.
It's just strange, and could be indicative of her lack of knowledge of the city and its overwhelming transportation system. She may have had false confidence at first and then realized...she couldn't take a train from Port Authority.
She tried for a ride from both Will and Missy and when that failed maybe she asked someone "how to get to Amtrak (Penn Station) or which train to Metro North etc. Maybe she asked someone that worked for the MTA or a transit Cop or something.
A note on walking around Manhattan late at night. The area that she was in is definitely lit up and busier than most so no it wouldn't be scary per se. But the later it gets the weirder it gets as well. As a woman in NYC, I don't care where I am, it is always unsettling to walk around alone really late at night, as is the subway platform (you'd be surprised how unpopulated it can get late night). And that area is just shady in general, with so much transient traffic in general. Port Authority and Penn Station are NOTORIOUS places for Pimps and traffikers to prey on runaways, unsuspecting girls new to the city, or girls already in "the life." It's pretty shocking really.
If you read the book "Girls Like Us" by Rachael Lloyd she gets into it. Rachael is the founder of a nonprofit here in NYC called GEMS (Girls Education and Mentoring Services) and works to help girls in the life to get out of it. Many die so young and are very often victims of severe abuse from pimps and johns alike.
She fights these issues often, like fighting to get backpage and Craigs List taken down. And Raising awareness about Domestic Trafficking in the US (We tend to call it "teen prostitution") when many of these girls are not of the legal age to even consent to sex under law (here its 17) and are being bought and sold by grown men. In any event, her book is hard to put down and will inform anyone who reads it about the world that these girls/women navigate and how truly frightening it is.
I think the average age of entry into prostitution is 14 years old in the US and their average life expectancy is somewhere in their late twenties. Here is a link to it, I highly recommend it. Rachael was a victim of it herself and ended up being saved by someone when her pimp tried to kill her. She then overacheived, got her masters, and opened GEMS to help other girls.
Here is the book.
http://www.amazon.com/Girls-Like-Us-Fighting-Memoir/dp/0061582069
Craigs List and the internet have made things that much more rampant.