A laptop that can play Fallout 3 on ultra even at 720p for $300? Easy... none.
Oups. Well, again, I don't think that Fallout 3 will run on anything else than XP or Vista. From a purely technical point of view, I'd assume most IGP from Intel COULD run it, but it's pretty much a dead game at this point.
Fallout 3 can be played on Win 7 and Win 8.1. I never had a compatibility problem running Fallout 3 with those OSes. However...
Fallout 3 and Oblivion (both use the same Gamebryo game engine) can be played using Intel HD graphics, but the game engine does not recognize integrated graphics (at least for Intel) to be valid graphic cores without a workaround. You need a d3d9.dll file that basically tricks the game into thinking the Intel HD graphics core is a dedicated GPU and will allow you to play the game. It can be obtained from the following link.
http://www.nexusmods.com/fallout3/mods/20371/?
The d3d9.dll file was originally created for the "Oldblivion" project. Oblivion was the first DirectX 9 game to be released. At the time a lot of people only had DirectX 8 GPUs. The d3d9.dll tricked the game engine into thinking the DX 8 GPU was a DX 9 GPU and allowed the game to run. Oblivion was able to run, but it kinda look bad from screenshots that I've seen because DX8 GPUs lacked features which DX 9 GPUs had.
For $340 the following Asus laptop is pretty much the best you can hope for..Spending less will basically get you an entry level laptop with very budget oriented CPUs with integrated graphics that are basically good enough to watch movie and play very simple games. The are the same CPUs that are used in Windows tablets. The laptop has a Core i3-5010u CPU (2.1GHz) with the Intel HD 5500 and a 1080p screen, but you should play the game at either 720p or 900p resolution. You also need to edit the Fallout3.pref file to disable all shadows which can improve performance by at least 6 FPS.
https://www.amazon.com/Asus-F555LA-AB31-15-6-Inch-i3-5010U-Processor/dp/B011KFQASE/ref=sr_1_1?ie=UTF8&qid=1475157343&sr=8-1&keywords=asus+core+i3-5010u
I actually installed both Fallout 3 and Skyrim on my laptop with Core i5-4200u and Intel HD 4400 graphics core (disabled the dedicated GPU) and Windows 8.1 since a friend of mine wanted to play those games on a laptop with similar specs, but no dedicated GPU. Fallout 3 generally ran better since it was an older game, but both games had similar minimum FPS performance. At 1600x900 resolution and some custom settings to a good balance between image quality (mostly a mix of low and medium) and performance, I was able to get something like 30 FPS - 60 FPS for Fallout 3 and 30 FPS - 45 FPS for Skyrim. This was during the summer of 2015.
Skyrim does support Intel HD graphic so no workaround is necessary.