Alaska is farther from the Lower 48 than most people are aware. And it is not off the west coast of Mexico, despite being placed there on some maps!
There is no direct rail link to Alaska exactly, so you either have to fly, drive, or sail part of the way. That said, Alaska Railroad does have interchange with CN and BNSF via barges that operate between Whittier, AK, and Prince Rupert, BC (CN), and Seattle, WA (BNSF). I think Prince Rupert is as close as you can get via passenger train. Conveniently, the Alaska state ferry system also serves Prince Rupert.
Driving up from the Lower 48 can vary widely on time depending how much sightseeing is done along the way. I drove once, but my main motivation was to simply get to Anchorage as quickly as possible. Driving about 14 hours per day, it took four days. That was with good weather the whole way and no vehicle problems. Both of those can add days to the trip.
Alaska Railroad itself is alone, and has no direct rail link to any other railroad. The barge connection is the closest thing we have to a connection to the outside. ARR operates year round and carries both freight and passengers. The 360 mile route between Anchorage and Fairbanks is where most of the traffic lies, but there are regular trains to Whittier and Seward as well, both south of Anchorage. There are also branch lines to North Pole and Palmer which see some usage. While the railroad operates year round, there is a pretty significant traffic drop every fall since the majority of the passengers are tourists. Many locals use the railroad too, but the number of tourists that visit Alaska in a year outnumber the number of Alaskans by nearly two to one.