So as it turns, out, a certain method of upscaling (the same method used for Vanilla Loading Screens HD, which that page explains)  works really nicely on most of the posters found in the game. The result is a crisp image without any of the ugliness caused by sharpening. Extra work was needed to keep the alpha (transparency) intact.

(Screenshots would be nice as I've completely forgotten where some of these are.)

Note: I've had mixed results with the graffiti, and for sure none of the DLC graffiti turned out correctly. If any of these textures end up actually looking worse in-game and I didn't notice, please let me know, and feel free to delete the offending textures!

0.9 Coming Soon:
Much better versions of the NCR propaganda posters

0.8:
urbanposters03, deleted in previous version

0.7:
Great Khan posters and flag
DLC graffiti in the Mojave
Dead Money Graffiti
Lettering in The Sink

New for version 0.6:
All of those posters that were reused from FO3
Nellis mural and sign
KEEP OUT and hobo graffiti
Eyechart (using the original font!)
Raider graffiti from FO3
OWB chalk drawings (not the one on Mobius' floor yet)
Logos on the B29, train engines, and disposal trucks
Chinese posters from FO3 that I don't think are used at all but now they look really nice

Included textures:

Followers Flag
General wasteland graffiti
Ulysses' graffiti - I fixed the alpha for this one, the transparency was misaligned
Hoover Dam graffiti
Text on graves
NCR Posters
Mr. House portrait
Rorchach tests
Poseidon Energy posters
Vault 11 posters
Repcon posters
Sierra Madre posters
Sunset Sarsparilla posters

Dead Money:
Vera's photo
Elijah's projection
Cosmic Knife poster and kiosk
Movie poster

Honest Hearts:
Zion National Park posters

Old World Blues:
Blueprints
Cosmic Knife poster (again)
Movie poster (again)
Ralphie poster
Repconn posters
Securitron posters
Sierra Madre poster
Dean Domino poster
Think Tank photo

Lonesome Road:
Ralphie Poster (again)
Red Glare posters
That government seal on the floor in the silo

To do:
Figure out a good balance of compression to use for the files with transparencies
Learn how to make vectors when working with 5 pixels
Either get permission to use Acoraito's weathered version of the few posters Bethesda released the high res art of, or learn to weather (the latter could be going better)