Mac Mini trying to install the latest iTune v12.x

The only real solution is to install a newer operating system version. Either that or learn how to use the iPod through iCloud which would likely also require a newer OSX version than 10.5.8 on the computer anyway. So I guess no workarounds at all for your current OSX.

iOS9 on a mobile device requires iTunes 12.3 or higher which in turn requires a computer running OSX 10.8.5 or higher. Software Update only checks for updates for the current system version you are running but that itself may be out of date. It may or may not be possible to upgrade your computer to the required system. Find your computer model on the http://www.everymac.com web site and near the bottom of the system specification section it will say what OS versions it is capable of running. If you cannot run a newer system you will not be able to sync this phone to your present computer. If it can run 10.8.5 or higher you can either buy a download code for OSX 10.8 Mountain Lion from Apple online, or you can try installing the free El Capitan OSX 10.11. El Capitan may run slower on older machines and require additional RAM purchase (4 GB is a realistic minimum and many recommend 8GB). Doing a big jump in system versions is also more likely to affect old software.

To do any later OSX installation will first require Snow Leopard:

1) Make sure your computer meets Snow Leopard technical specifications - http://support.apple.com/kb/SP575 This mostly means making sure it is an Intel Mac and has at least 1GB RAM (at least 2 GB is recommended). Go to the Apple in the upper left corner of any window to "About This Mac". To check RAM continue to "More Info...". It should say you have at least 1GB of "memory".

2) Buy a Snow Leopard disc from Apple online

http://store.apple.com/us/product/MC573/mac-os-x-106-snow-leopard - USA

http://store.apple.com/uk/product/MC573/mac-os-x-106-snow-leopard - UK

or telephone Apple. In the USA: (800) 676-2775 (contact information for other countries: http://support.apple.com/kb/HE57)

Alternatively you can buy from online retailers but anticipate paying much more than the original price which is the price still charged by Apple

3) Backing up your computer before upgrading is always a good idea.

4) After installing from the DVD, update your Snow Leopard installation using Software Update or manually downloading and installing the Mac OS X 10.6.8 Update Combo v.1.1 - http://support.apple.com/kb/DL1399

It is advisable to use Software Update at least once so you get special updates such as the "Mac App Store Update for OS X Snow Leopard" - https://support.apple.com/HT205702

If you are using old (PPC generation) software you may want to install Rosetta as well which I believe is an optional installation, but if you have any PPC generation software that has been running under OSX 10.6 it will not be supported under any later system versions.

Snow leopard to el capitan, will it cause my macbook to be slow? - https://discussions.apple.com/thread/7412959

Mountain Lion 10.8 purchase link U.S.A. - http://www.apple.com/shop/product/D6377Z/A/os-x-mountain-lion

Mountain Lion 10.8 purchase link U.K. - http://www.apple.com/uk/shop/product/D6377ZM/A/os-x-mountain-lion

Current OSX general upgrade information, including system requirements - http://www.apple.com/osx/how-to-upgrade/

Beware any future iOS upgrades if you run older equipment. You normally do not have to upgrade iOS just because Apple brings out something newer, and a newer iOS may require a newer OSX which will require a newer computer and a thousand dollars in software upgrades, and so on and so on. Because your computer is reaching the end of its Apple-perceived lifespan you need to be very careful before installing any new software or buying any new equipment as this may set off a chain of events. iOS10 is likely to be released pretty soon now. I don't know what its requirements are but it may bump things up even further (maybe you will have to install El Capitan?). Your computer use is about to undergo something of a shock with you jumping from 10.5 to whatever and you don't want to accidentally lock yourself into having to spend thousands on a technology upgrade just because you upgraded iOS on a device.