I just want to know if I can buy a Steam game from my phone, and then just download it on my PC?
Yes, you can buy a game now and download it to your laptop later.
You can either use web version of steam store - https://store.steampowered.com or their mobile app - https://store.steampowered.com/mobile . Just login into your account and make purchases as usual. Just don't forget to check if the game is available on platform your laptop is using (Linux, Windows or MacOS).