Is there a way to make the midi arpeggiator to sync with the tempo of the song?

Are you talking about the Midi Arpeggiator plugin or the built in Arp on every Midi track?
 
I've never used the plugin but the track Arp defaults to be in sync

I think it depends on your input, being in time, and  in sync, not the arpeggiator itself. The arpergiatted notes created will be based on the tempo. If you play late, the first arppegiatted note would have to be shortened, for the following notes to be in sync.
 
If you input quantise, what you play is forced into sync and allows the arppegotor to then be in sync.
 
 
That was a really bad explanation. Needs pictures really