Re: Battery Tunnel, yes, there absolutely is a ramp to head north, it actually digs down in a short tunnel to do it.
Re: skyline opportunities, besides the two mentioned:
1) First Mountain. You can park at Highlawn Pavilion off of Eagle Rock Ave. in West Orange and get a great (but distant) view of the entire city and bridges.
2) Union City. There are great views near the southern end of CR 505, which would be along JFK Boulevard East (not regular JFK Blvd., which is a separate road) in the area of 60th St. roughly. (I see a place called "Liberty Skyline" on the CR 505 incline, called "Anthony M. Defino Way" on Google Maps.) Parking's a bit tight in that area though, because I'm not sure if permits are required on side streets.
3) Atlantic Highlands. This would be anywhere along the coast north of NJ 36 from Union Beach east, or even up Sandy Hook (as long as it's before May, access is free to Fort Hancock).
I don't think Brooklyn is going to offer you as good of a skyline. In my experience, there is too narrow a field of view coupled with seeing the "back side" of many buildings. What you can get there is a view of all the great bridges across the East River. The only good view I've ever gotten from Brooklyn is from the Amtrak train, which rides high above the city. You might consider trying Brooklyn Heights, which is the top of the quad-deck area of I-278/BQE north of the Battery Tunnel. Since I've never tried it, I don't know if parking is free or not.