Did you check to see if there is a timer in the bios for how long it displays the boot screen? Some motherboard bios settings have a value that usually is 5 seconds but it can be changed to boot past bois screen faster, say in 3 seconds or 1 second.
That processor supports up to 8GB of RAM. The processor sadly is soldered to the motherboard (as is most these days) so you're stuck with it. An ssd would speed it up considerably as well.
Try removing nonessential hardware such as CD drive and HDD. Also disconnect keyboard and touchpad. Do these one at a time until you have them all disabled then if it boots try connecting them one at a time to find the fault. If this doesn't work I do believe you have a processor power routing issue.
Does it have a cooling fan and is it working? Could be overheating. If you are proficient enough and have spare parts you can swap out the RAM and try running the machine without unnecessary hardware (CD drive, HDD) to see if it stays on longer than 2 mins.
Check BIOS settings and try disabling secure boot. Also you can check the Sata Hard Disk Mode setting in the bios and set it to AHCI mode and not IDE. Hope that helps.