All tourist activities are winding down in the next wk or 2. In a couple of weeks, snow is expected in Anchorage and even earlier (like now) in the high country. Train should still be running. But many tour companies will close the door for winter.
Anchorage the city itself doesn't have much to offer - it's the surrounding wilderness that's worth visiting. 4 days are NOT enough, as Alaska the state is about the size of 1/4 of US lower 48 states. Any direction you go would be most likely hours of driving.
A few suggestions in the Anchorage vicinity for your 4 days:
-In Anchorage, go up to Flat Top Mountain to get an overlook of the city, Turn Again Arm and Cook Inlet and on a clear day you might be lucky enough to see Mt Mckinly. Sunset time is the best. The float plane base across from the airport, being the largest in the world is also unique and very Alaskan.
-drive to Hatcher Pass to visit the Independence Mines and it is a nice scenic day trip from Anchorage. However, the mine won't be open when you go.
I do expect snow for sure by the time you go.
-Drive the Seward highway to Portage Glacier, Girdwood and maybe go to Whittier, and take a boat tour of tidal wave glaciers from Whittier if they are still open (Call them first; I don't think they will be open late Sept). You can make multiple stops along Seward Highway. Do check the Whittier tunnel schedule if you plan to go to Whittier.
-Drive the Parks hwy to Denali but that will be a very long day trip. I know in may before the Park shuttles start running, you can drive your car into the park. Don't know if they let you do that in Sept. Plus you will driving on dirt road, violating your rental car contract. Do not recommend it. If the weather is extremely clear, you could drive to Denali State Park and get a very nice view of McKinley. This is only if Mckinley is visible (you have 20% chance of seeing McKinley due to the weather)
-drive Seaward highway (you can go all the way down to Seward but your time may be too short to do that). If you get to Seward, visit Exit Glacier and take a boat tour of Kenai Fjord NP to see tidal wave glaciers (however, tours are probably closed by end of Sept)
-Drive to Matanuska Glacier (2 hours approximate on Glen hwy). You can go further to connect to Richardson hwy south to Valdez but I don't think you have time.