I had a quick look at the manual:
First step is to connect the device to your PC using USB.
The Korg should show up as MIDI device in Ardour’s Menu > Window > MIDI connections. – if not, check Window > Audio/MIDI setup – what OS are you using?
In the MIDI connections window, connect “Ardour/Misc” Sync > MIDI-clock out to “Hardware” system:… (midi port corresponding to the Korg)
Last but not least: in Ardour’s Preferences. Sync > MIDI > Enable Mclk (MIDI beat clock) generator.
The Korg must be set to use “Auto (USB)” as clock-source, see page 41 in the manual.
Press “play” in ardour, and the Minilogue should pick up the BPM.
To record MIDI, create a MIDI track, it should auto-connect to first available MIDI port, if the Korg is not the only MIDI device, you may have to manually change this.
You may also want to disable the preference: “MIDI” -> “MIDI input follows track selection” (to avoid accidental disconnects).
Anyway, I’ll keep my fingers crossed. If it doesn’t work, it’ll be helpful to know what you’ve tried and where you’re stuck.