Will I have a hard time driving from NY to Alaska (Prudhoe Bay) in a 2004 Dodge Neon during the summer?
As long as you go in the summer, you won't have any problem driving to Alaska. People do in on motorcycles all the time.
If you can make it through the Rockies, Alaska will be a piece of cake.
The drive is beautiful. You can camp along the way, or stay in hotels. If you don't want to drive the entire way back, you can take your car south on the Alaska Marine Highway System from either Haines or Skagway to Bellingham, Washington. You can sleep on deck in a sleeping bag, or get a ferry stateroom. Best to get reservations, if you can plan that far ahead.
You can drive north on the Dalton Highway, but it's a truck route and parts are very rough. Most people only drive part of the way, just past the Yukon River. High risk of flat tires and rocks in the windshield. Not to mention being passed by large tractor-trailer rigs. Once you get to the oil fields, you have to stop, so you would not be able to drive all the way to the ocean without permission from the oil companies.
Alaska is light until late at night, so you can drive pretty much any time. That said, some people drive when they are tired, fall asleep at the wheel and then run into someone. Since you are in a Neon and most drivers will have motorhomes, trucks and SUV's, you will want to be extra careful to watch out for someone crossing the center line.
Distances in Alaska are deceiving, because the roads are not freeways, so don't expect to make the daily miles that you may have done on your other road trips. You won't be able to drive 70 mph. Plus, you will want to stop and look around, photograph moose, etc.
Buy a copy of the Milepost, which is a guide to the Alaska Highway and the roads in Alaska. You can find a copy on Amazon.
Make sure you have a good spare tire, not one of those tiny ones that's good for about 20 miles. It'll be longer than 20 miles between gas stations. Make sure you have a good jack and jumper cables. Bring some extra motor oil, belts, etc. for your car and the repair manual (you can buy on Amazon) so a roadside mechanic can help you if you break down. Join AAA so you won't have to pay towing. A long tow will easily cost you $700.
Bring food and drink in your car.
Count on your car radio having some dead spots along the way. Same with some cell phones.
Be prepared to cross US and Canadian borders. Make sure you have the necessary papers for you and your car, pets, etc. and that you know what you CAN'T bring across borders.
If you plan to drive gravel roads, think about buying the deflectors that are supposed to keep the rocks off your headlights and windshield. Rock chips in the windshield can be fixed at a Novus place, if they are not too big. Get them fixed asap before they grow into cracks. Maybe get a car "bra", too?
Take a look at an Alaskan travel guide, such as the Lonely Planet guide, to plan your itinerary. Stop at visitors' centers. They have maps, etc. and often the visitors guides have coupons for discounts and free stuff.
Be prepared to meet some wonderful people and see some amazing sights. It's a trip of a lifetime.
One possible itinerary would be to go north to Fairbanks and places north, then south to Denali National Park, then to Anchorage and the towns on the Kenai Peninsula, then back to Anchorage and on the Glenn Highway towards Glenallen and possibly the Wrangell St. Elias area, then north to Tok and back onto the highway to Whitehorse and down to Skagway to catch the ferry.
Gas in Fairbanks right now is over $2.50/gallon.