Yes, can record internally by midi sends but you cant send a track through the arp. It's either the pads or external midi that goes through the arp. In order to get a track already recorded to use the arp you have to send the midi out through a cable and then back into the midi in port of the MPC. Then that midi will get processed by the Arp and you can record. But there is a small noticeable delay and it will be perfectly off grid so you'll just have to select all notes and nudge it back to start to align.
You dont need an external controller just one midi cable in to out. And make sure your track settings are not causing a midi loop. Although aftertouch and pad velocity seems to get stuck in a midi loop anyway note on note off duration works fine. I only recommend doing the midi cable looping when you are specifically wanting to route through the arp. Does seem a bit silly that a track cant use the arp internally though.