Hopefully, someone who has been there can provide you much more detailed information from a firsthand experience.
Unfortunately, we've not been there yet, so the information I'm sending is from a client of ours based on your questions.
This is one of those ports that was build by Carnival Corp. and like all of these types of ports, such as Mahogany Bay, Falmouth, and Costa Maya, they were designed to try and keep you within the gated area, so you'll spend your money with them. Yes, there are excursions that will take you to Puerto Plata for sightseeing purposes, but you can do it on your own, if you prefer.
She said Puerto Plata is about a 20 minute taxi ride from the pier, so definitely not something you'd want to walk. There is a taxi stand, but evidently they're not cheap - around $35 each way. She said she heard others saying that getting a taxi outside the gated area was cheaper than getting those inside of Amber Cove. (Don't know if that's true or not, but we've personally found this to be true in several other similar ports.)
There is a National/Alamo car rental office within Amber Cove, so you could check that out if you're so inclined.
As for shore excursions, I do know that there are some independent companies offering a few, but evidently there are not alot being offered, yet, since it's a relatively new port. So, most people are either doing their own thing or booking excursions directly through the cruise lines. Again, as with most of these types of ports, I'm sure there are others available outside the gated area, if you'd like to give them a try, which is what we prefer to do in such cases.
I'm sure others will have some useful input for you, but please come back and provide us some of your personal observations and experience after your cruise. We'd love to hear all about it!