I have a first generation Apple TV.
The big white one?

I lost my remote.
The remote for the 2nd and 3rd generation (aluminum) will work with any model Apple TV so far.

I was using a Remote app, but it stopped working.
Updates to the app have abandoned the oldest model. The original app has been further developed as two separate apps now: iTunes Remote app and Apple TV Remote app (for 2nd generation or later).
I ordered a replacement remote but I need to be able to turn on the Apple TV to pair it.
Reconnecting power should turn on the Apple TV. You will need to unlink the old remote, though:
You can use any Apple Remote to unpair (unlink) Apple TV from a remote. If you lose the Apple Remote that you paired with Apple TV, a replacement remote won’t work until you unpair Apple TV from the lost remote. If your Apple TV status light (LED) blinks 3 times in quick succession when you press keys on the remote but fails to respond on-screen, try these steps to unpair your remote:
- On an aluminum Apple Remote, press and hold the Menu and Left buttons for 6 seconds.
- On an earlier, white Apple Remote, press and hold the Menu and Previous/Rewind buttons for 6 seconds.
When you’ve successfully unpaired a remote, Apple TV displays a broken chainlink symbol
above an icon of a remote. You can now pair Apple TV with a different remote.
If your Apple TV remote isn’t working - Apple Support
I cannot read the serial number on the bottom the Apple TV
I’m sure you won’t need the serial number for this. Once you have it working, you can look it up in Settings, though, and copy that for your administration.