Q: My computer is asking whether to set up my not so new iPhone as new, or to restore from a very old backup...

Purchases from iTunes, or the App Store?

One thing that you will have to do for certain is to transfer the purchases that you made on the phone into your iTunes library so that the two devices will have matching content or so that iTunes will recognize the phone as belonging to the iTunes library on the Mac.

Redownload or transfer your iTunes Store purchases from an iPhone, iPad or iPod to a computer - Apple Support

Before you do this, launch iTunes (without connecting the phone) and go to iTunes>Preferences>Devices and check the box at the bottom of the window that says Prevent iPods, iPhones and iPads from syncing automatically. Click OK and quit iTunes. Now connect the iPad and launch iTunes on your own. When you chose to not auto sync, that disables iTunes form auto launching when you connect the phone, then follow the instructions in the article about transferring purchases. After all purchases have been transferred, try syncing again and see if you still get that message.

As  long as all of your iPhone and iTunes content was purchased from iTunes, or the App Store, this may be all that you need to do. If you have content in your iTunes library that came from other sources such as music imported from CD, this process may work a little differently. You might have to select all of the non purchased content on your own to sync to the phone.

I assume that you did select the phone in the devices tab/icon at the top of the iTunes screen or you wouldn't see restore backup. Are you signed into the same Apple ID on the phone as you are on your computer?

Have you tried disconnecting the phone, resetting it, closing iTunes, rebooting your computer and then try again. Reset the phone by holding down on the sleep and home buttons at the same time for about 10-15 seconds until the Apple logo appears on the screen.