The file that contains the patches has a readme with a small tutorial on how to use them, it's really easy to understand so it shouldn't be a problem to install the patch. Just remember you need to reset the P's once you've installed the third patch
The patches act like destinations, so you put them on the Tamagotchi (in the right order, 1st patch first, then second, etc.) and choose them on the door menu, your Tama will get in a tram and then will ask you if you wanna install the patch, it takes some minutes for every patch to install.
As for ir mobile devices, most of them are cheaper than 25$ if you buy them used, I would recommend going to a second hand shop and check the old phones they have, any phone that has IR and can send files will work, so don't worry about that
Some people also use IR USBs, but those tend to be more expensive/harder to find and do not always work