It's not a straightforward answer. It's bloody complicated. But do not despair.
The 23 inch Apple display in the wide Lucite silver-white frame has an ADC connector that accepts 28 Volt power at up to 4 Amps from the ADC connector. If you are disconnecting from your ADC computer, you will need a Power Supply. The Apple DVI to ADC converter is such a supply. (It has an AC power cord.) It is readily available on the used market. Once installed, you have a DVI display with a power brick in the cable.
Thunderbolt connector is display-compatible to Mini DisplayPort. You can force every Mini DisplayPort (including Thunderbolt) into "Single-Link DVI compatibility mode" by using the correct adapter. Buy a Mini DisplayPort to Single-Link DVI adapter from the Apple Store. If there is any trouble, take your all-Apple equipment to the store and ask them to fix it.
For the 23 inch display, you do not need the much more expensive Mini DisplayPort to Dual-Link DVI converter, because the display you are using is not over 1920 wide, and can therefore use Single-Link DVI. IF you were hooking this up to a Mac Pro (tower), and wanted to use it on a card with more than one Mini DisplayPort, the rules are MUCH more complicated.