North to Fairbanks is a long and mostly boring drive. And to see Denali you really need a full day on site, since the interior of the park (where the views are) is only accessed by shuttle bus, and takes most of a day just to get in and back to the visitor center. Counting the drive both ways, it would really consume most of the time you have available, and there's no guarantee that the mountain will be visible on the days you're there. And, with short notice like this, it's doubtful if you'll be able to find accommodations near the park anyway.
Going south is the preferred route. If it was me (and not knowing how far you're coming on arrival day, or from what time zone) I'd book someplace in Seward for the night of the 1st, and just drive directly there from the airport (around 2 1/4 hours.) Don't worry about driving after dark - it's not possible in early July. If you can't find something in Seward, or else are worried about being too tired, look for accommodations in Girdwood, around 40 min. south of the city on the Seward Highway.
Then on the 2nd, take a cruise to Kenai Fjords National Park. It's sort of "instant Alaska" - spectacular scenery, wildlife, etc. And spend that night in Seward.
Leave Seward on the 3rd, because that night the town will be inundated with runners planning to do the Mount Marathon run/scramble/attempted suicide race on the 4th. Again, you might have difficulty getting accommodation in Seward due to runners and especially due to boat people getting ready to leave Seward on cruises the next day. If Seward is plugged, look for space in Cooper Landing or Moose Pass, 20-30 min. up the road from Seward.
On the 3rd and 4th, return to Anchorage and explore the city - walk or bike on the coastal trail, or, if your wallet is up to it - take a never-to-be-forgotten flightseeing tour on a floatplane from Lake Hood. You simply can't get a feel for Alaska from the ground - get up in the air. Don't miss the Native Heritage Center, maybe hit the museum. Have a burger at the Arctic Roadrunner. Hike up to Flat Top.
Starboard window arriving, port departing.