Last question first; Yes, you will need a "bootable" installer which will circumvent the need for a functional drive in the Mac in question.
As for instructions, it's relatively simple.
First you need to download a macOS installer from Apple (App Store) that suits the Mac in question. You don't say what Mac model or year you have so what macOS were you trying to update to?
Assuming you can find it on the App Store you will need to download it onto your "newer" Mac. You may get a message to say it cannot be installed on the newer Mac, "Are you sure you want to download it?" Just confirm yes.
Now you will need a USB Thumb Drive of 20GB + in size. NOTE; it will be erased by this process.
Also, I prefer Instal Disk Creator available free from here;
https://macdaddy.io/?utm_source=app_idc
Once you have downloaded the macOS, plug in the USB stick, startup Instal Disk Creator and it will ask the location of the macOS installer (source), point it to the Applications Folder unless you put it somewhere else, it will find it there.
You will need to select the "target" being your USB Thumb drive.
When you press start Instal Disk Creator will erase the USB drive and begin the process of making the installer bootable. It can take a while.
When complete eject the USB drive and plug it into your old Mac which should be powered off.
Turn the Mac on holding the Option Key until you see "boot" options, choose the USB.
Once your old Mac starts up you will see what looks like the recovery partition.
Now, at this point I would be inclined to erase the old Mac's drive using Disk Utility but you may have data on this that you want to save so, just choose instal macOS ..... from the recovery screen.
That should work, if so all of your data should also be there.
If you don't need the data or you have it backed up somewhere else you can open Disk Utility from the Menu bar, select the internal drive at the media level (the manufacturer's name like eg Samsung 500GB HDD) and click Erase.
You will need to Name the drive = Macintosh HD. Format the drive = either MacOS Extended (Journalled) for old devices or APFS for more recent devices and choose GUID for Mapping.
Now Quit Disk Utility and return to the recovery screen and click Instal MacOS ....
When complete, reboot with the option key again, this time choose the macOS you installed.
The result of this will be a device that starts up like a brand new device with the Setup Assistant.
You will need to enter all of your user data as you did when it was new; Location, Language, User Name, Admin Password, email, Apple ID ect.
Just like a new device you will have to option of Restoring data from a backup as part of the setup.
The above is what a repair shop would do with the advantage of them most likely having the bootable installers for a number of macOS already. For them it's just a matter of plug in, erase, install.
