I'm writing a story following a character that's part of a culture that really loosely based on the Viking era. I was working on a scene where he's bringing his ship into port and I realized I don't really know what that would involve. I've done a bit of research and can't too too much about this specific thing.
I'm wondering if anyone could give me even a brief idea of what that process would look like? Would they row their way in, for instance? Would they use the sail? That sort of stuff.
Your best bet is to look at what's being done with reconstructed vessels at the Viking Ship Museum in Roskilde, especially with their longship, the Sea Stallion of Glendalough (or, in Danish, Havhingsten fra Glendalough). The Sea Stallion is reconstructed based on an archaeological find known as Skuldelev 2, which was built in Ireland in 1042 and scuttled to restrict access to the Roskilde harbor (we don't know why) around 1070. If you look at the pictures of Skuldelev 2, you'll see how fragmentary it is, which is pretty typical of archaeological finds. The experimental archaeologists (what a cool job!) who reconstructed this thing needed to make a lot of educated guesses, informed by things like medieval art and modern trial and error. The biggest question marks revolve around the rigging. We simply have no surviving traces of what this looked like, though researchers made good deductions based on how the hulls of these ships were designed and what we know of later medieval rigging traditions.
Those hesitations aside, there's tons of footage out there to help you understand how we think these things were handled. The Viking Ship Museum has a collection of short videos available, mostly in Danish, though some might still be interesting to non-speakers. The ship guild (i.e. the people who pay for a chance to be part of the crew) has a smaller selection of videos, although I'd especially point to the last two—a 3D reconstruction so you can see the insides and a 2-minute greatest hits of the Sea Stallion's voyages; both are typically on display at the museum itself. Finally, there's a superb BBC special on a voyage the Sea Stallion made from Roskilde to Norway to Scotland to Dublin back in 2007. Apparently arriving in port isn't eye-catching footage, but they do show the slow sail into Dublin itself beginning at 1:25:16. In general, you should be aware of a few things:
You needed to drop your sail. Trying to stop with a sail pulling your boat is difficult and dangerous. You might lose an anchor, damage a sail, or worse. On more modern sailing ships, this is often done by pulling the sail (the cloth) up to the yard (the cross beam). You'll see on the reconstructions, however, that they typically lower the yard down to the ship and then tie the sail to it.
You'd probably have to row. Longships had big sails that were difficult to handle, so unless you wanted to risk the aforesaid damage, you'd want the sail down sooner rather than later. This might require the whole crew rowing for a while, but modern-day crews usually break up these responsibilities. The Sea Stallion is divided into three "rooms" (bow, middle, and aft; each has different jobs when handling the sail, so these are a somewhat natural division); the two sides (left or port and right or starboard); and IIRC two "watches" (who's on or off duty). This gives the skipper significant control as he maneuvers his ship into port, perhaps having the starboard bow A watch rowing ahead while the port stern B watch rows in reverse so that the ship rotates clockwise around its center.
You might use ropes. The modern Sea Stallion is kept in the most protected part of the harbor since it's the prize of the Viking Ship Museum's fleet. I expect similar considerations guided the docking of Viking-Age ships. But to get to this spot, the Sea Stallion needs to negotiate some narrow passages (including a walking bridge) and turn some tight angles. The crew tosses ropes back and forth, fastening them either to the ship or to posts on the shore, so that they can carefully pull the boat in a controlled direction. This is pretty slow work, so it probably would have only been done in a crowded harbor, and those were pretty rare in the Viking Age.
At any rate, you can see all these things at work, if only briefly, in the BBC special linked above, beginning at that second link of 1:25:16. The Sea Stallion sails up the main channel, drops sails and rows for a bit, with only parts of the crew rowing in some shots, and then it gets pulled in to the moorings with ropes. There's no fancy maneuvers, but it's enough to give you the full idea.