Sure, it's possible to run it outside, but unlike a TV wire it'll need to be protected from the weather and physical damage, so you'll have to run in conduit and it'll be an eyesore.
In general, it's usually easiest to fish wires in an outside wall in old houses.

The house may be balloon framed and extremely easy to fish cables in the wall.
If you wish to use cable on the exterior, Type UF that is Sunlight resistant can be used with protection, usually a piece of EMT with suitable end fittings per code, for a height of 8 feet above the ground. Above that, no protection is required. Makes for a crappy-looking installation but is quick, cheap and meets code.

Thank you both. We are not really concerned about the looks as there are already a bunch of other wires rinning down the exterior in the same location so one more will not make a difference! We are going to try and find a way to fish them inside first as suggested but just wanted to make sure we had a back-up plan incase we ran into trouble (which is pretty normal with an old house!). Thank you again for the speedy responses!

Even with plaster and lath (lathe is a device for turning wood) it's probably not hard to run things interior. Small holes can be cut and repaired. Even if platform framed, you can drill the plates. If we're talking about a masonry exterior, there's often plenty of voids to pull through, other places that are handy are along the runs for chimneys, etc...where there are often gaps in the framing.

While others have pointed out the legality of pulling things on the exterior, it's a pretty hideous thing to do to an old house.

Find a wall on the first floor that has another wall above on the second floor.

Cut out the plaster in the stud bay and run your lines and miltuple conduits in the bay from basement to attic.

Repair the wall section as well as you wish.

Anything from drywall up to plaster on expanded metal lath (3 coat plaster) or gypsum lath (2 coat plaster).

If you need any other lines (phone, plumbing, cable TV, thermostat, etc.) this is your chance to run them.

You need to decide if you can tolerate a sub panel on the second floor somewhere.

You can then run four larger lines for the sub-panel directly or to the attic and then drop them back to the sub-panel (panels are not allowed in clothes closets).

Behind a door to a room is often a good hiding place.
When the door is open the panel is not visible.

Years ago I ran a power wire from the attic to the basement in my lath and plaster walled house, going through the small space beside the main plumbing stack. Don't know if any NEC rules were violated but it worked for me. Or maybe alongside the chimney, as ronnatalie suggested. All the best. Bernie