Connecting a Mac Monitor to an old (2003) Mac Mini

You say you have an "old (2003) Mac mini." There were no Mac minis in 2003. The very first Mac mini came out in January 2005. That one had a DVI connector and supported a single monitor with a resolution of up to 1920x1200 pixels (DVI) or 1920x1080 pixels (VGA).

Mac mini (original) - Technical Specifications - Apple Support

There is no way this Mac mini can drive an 27" Apple Thunderbolt Display (model A1407). If you have incorrectly identified your display and you actually have a 27" LED Cinema Display (model A1316), that monitor would still have a higher resolution (2560x1440 pixels) than your Mac mini can support, and any sort of adapted, digitally zoomed output would look terrible.

I would suggest getting a 24" 1920x1080 or 24" 1920x1200 display that has a HDMI input (among other inputs) – and using it in combination with a single-link-DVI to HDMI adapter. (You could also run the signal from the Mac mini into a 4K display or UHD TV that has the ability to display 1080p input, if you happened to have a 4K display with a spare input lying around.)

The Apple Thunderbolt Display has a TB port & requires such on the Comuter's end...

Compatibility[edit]

The Thunderbolt Display drops compatibility with all previous standards, including VGA, DVI, and DisplayPort.[8] It is not compatible with computers that do not have a Thunderbolt port, including pre-2011 Macs and the vast majority of desktop PCs. The 12-inch Retina MacBook and 2012 Mac Pro do not support Thunderbolt. The following Macs support the Thunderbolt Display without an adapter:

• MacBook Pro (2011 to 2015)
• MacBook Air (2011 to 2017)
• Mac Mini (2011 to 2014)
• iMac (2011 to 2015)
• Mac Pro (2013)

Macs released after 2016 with Thunderbolt 3 and later, which uses a USB-C connector, are compatible using Apple's Thunderbolt 3-to-2 adapter.[9]

https://en.wikipedia.org/wiki/Apple_Thunderbolt_Display

An Apple Thunderbolt Display requires a Mac with a Thunderbolt port. There is no way to adapt any other display interface technology to it.

Jzapko13 wrote:

So the display need thunderbolt port

As I said, the Apple Thunderbolt display requires an Apple computer with a Thunderbolt port!

I have one and know that for a fact.

If you have a Mac Mini that is 2009 or newer, you could use the mini display port to hook up a 24" LED Cinema Display (model A1267) or a 27" LED Cinema Display if you like the look of the Thunderbolt display (model A1316). The 2009 models are also the only ones capable of dual-link DVI to power a 30" Cinema Display (model A1083). The older Mac Mini's are stuck with regular DVI and max resolution of 1920 x 1200

whats the price limit?

So the display need thunderbolt port