There is a quick and easy way to do this, no special tools, $10 of parts, 20 minutes work if all goes well.
Cut in an old work low voltage mounting bracket on either side of the wall, back to back
and use a "duckbill" cover for a finished appearance, (the one in the photo is an Arlington CE1)
All you need to do this is
- a level to make sure you position the bracket straight / square,
- a pencil to mark the outline of the bracket,
- a drywall saw to cut a hole for the bracket,
- a screwdriver to screw the bracket in, and screw the duckbill on,
- and optionally a studfinder to make sure you pick a clear hollow spot.
When you pick your spot, try to match the height of any nearby electrical receptacles. Try to stay well away from studs. You can use a studfinder, or look for the nails holding the base molding to the wall - those will be nailed into the studs. You can also turn off the breaker for a receptacle and take the cover plate off, you'll probably see the box is nailed to a stud, and you can go on the other side of that stud.
When you cut the hole for the bracket, make a horizontal cut first so you can adjust if you miscalculate and hit a stud. Don't plunge the saw in - take your time, use just a couple inches of the blade, just in case there's something behind there you don't want to hit.
edit After you cut in the first bracket, move any insulation and poke the drywall saw through to the other side, just enough to pierce the drywall so you can place the bracket on the other side of the wall.
If you stop using the pass-through some time in the future, you can just remove the duckbills and put blank covers on.
The tool that is made perfect for this is the type of unit pictured below:
(this picture courtesy of MyToolShed web site)
These oscillating multi-tools can accommodate a variety of blades that permit close cutting exactly like you need to do. Here is an example blade that is what I would use for this cutting job.
(this picture courtesy of Lowe's web site)
These are available from many manufacturers, both AC powered and battery powered. I went for years before getting one just a year or so ago and now use it for many jobs I used to struggle with. A particularly nice feature of the unit pictured is the LED light on the front of the unit to illuminate the cut area.
Best Answer
Command strips "cord bundlers" and a nice hunk of decoy fake Ivy or other imitation vining plant (unless there's enough light and willingness to care for to use a real plant - unlikely.)
Alternatively, fairy lights, etc.
Basically, it will be in plain sight any way you slice it, so might as well just make it disappear by having something more noticeable that makes it non-obvious (the decoy need not stop where the ethernet wire does, either, which can help.)