My Xbox won’t stay on for long and the power button doesn’t work

the power button is probably just overused and has broken. opening it and testing with a multimeter on continuity would help you troubleshoot that issue,

as for it shutting down.

try blowing it out with some air, that console is probably ATLEAST 10 years old. it should be opened up and cleaned out completely, new heatsink paste, dusted out with a brush and reassembled. that would be your best bet on why it starts and shuts down after a while.

if you try this and it still does then we can further troubleshoot the issue.