Thunderbolt Connection On Old Mini

That Mac Mini has 4 USB 2.0 ports, 1 Firewire 800 port, Bluetooth 4.0, and WiFi. The most recent version of macOS that it can run is High Sierra, so you would be using iTunes and doing iTunes-based synchronization.

Sync your iPhone, iPad, or iPod using your computer - Apple Support

Use iTunes to sync your iPhone, iPad, or iPod with your computer - Apple Support

Sounds like you need to do the initial synchronization over USB, though you might be able to do subsequent ones over WiFi.

The iPad Air (5th generation) has a USB-C port, and comes with a USB-C to USB-C charge cable. You're going to need

1. macOS El Capitain 10.11.6 through macOS High Sierra 10.13.6 on the Mini.
2. iTunes 12.8 on the Mini.
3. A USB-C (iPad Air) to USB-A (Mini end) connection using a cable (or cable + adapter combination) that is rated to carry data. (Some very cheap charging cables do not carry data or carry it only at USB 2.0 speeds.)

If you need to upgrade the operating system on the Mini, see this page:

How to download and install macOS - Apple Support

If you need iTunes 12.8, you can get it here:

iTunes - Apple

As for adapters, Apple sells a $20 USB-C to USB-A adapter. You could combine that with a USB-A (USB 3.0) to USB-A (USB 3.0) cable that has USB-A plugs on both ends. Or you could go looking on Amazon, where you will find many adapters and cables of various sorts and prices.

That port doesn't have the same connector as the Thunderbolt ports on the newer Macs. It is a Thunderbolt 1 port that uses a Mini DisplayPort connector – and that can double as a Mini DisplayPort.

Starting with version 3, Thunderbolt moved to the USB-C connector. USB-C is a "Swiss Army Knife" connector. It can carry USB data, USB Power Delivery, DisplayPort Alt mode, and Thunderbolt Alt mode signals. The TB3 & TB4 ports on newer Macs carry several types of signals, and select among them depending on what's plugged in.