How should I run Ethernet from downstairs to an upstairs PC in a rental?

I would try the indoor route first. Measure along the baseboards, corners and stairs rather than straight between the rooms, then buy a cable long enough for that path. Adhesive cable clips can keep it tidy without drilling, though check that the adhesive is suitable for the paint.

Keep the cable out of walkways and door hinges so it doesn't get tripped over or crushed. A 100-foot cable may work, but measure before ordering.

Outdoor-rated Ethernet exists, but ordinary indoor cable shouldn't be left outside. Look for a cable specifically rated for outdoor use, with resistance to moisture and sunlight. Some outdoor cable comes in bulk and needs connectors fitted, so check whether the one you buy has plugs already attached.

Windows can pinch a cable and leave gaps around the seal. In a rental, check with the owner before making any openings or alterations, and avoid a route that prevents the windows from closing securely.

Check whether there are coaxial cable outlets near both the modem and the PC. MoCA adapters can use suitable existing coax wiring to carry the network connection between rooms, so you might not need a cable up the stairs.

You'll need compatible adapters and connected coax outlets. Check the layout and equipment requirements before buying anything, since having outlets in both rooms doesn't necessarily mean they are connected in a usable way.

Powerline adapters are another option if a long cable isn't practical. They use the home's electrical wiring, but performance can vary a lot with the wiring and interference, so they're less predictable than a direct Ethernet cable.

I wouldn't assume a ventilation duct is a free cable route. Air-handling spaces can have specific cable and fire-code requirements, and a rental may have its own restrictions. Check with the owner and a qualified installer before considering it.