Am a little puzzled on how you have the same email account configured <twice> - that by itself is strange (to put it mildly).
Only thing I can suggest is to remove all instances of this account and re-add it. After doing that, make sure it's set to <Sync> under <Contacts> is set to <On> and tap on Sync to force an immediate sync.
Just out of curiosity - is this an account configured under two different <email clients> (1) with the mobile Outlook app and (2) with something else?
FWIW, I only have the Outlook.com account configured using the mobile Outlook app which has worked perfectly from the outset and just to be sure, before posting this response, made a change to a contact on my phone to see if got reflected on my PC - it did, almost instantaneously.
Did the same in reverse (change on PC) - wasn't reflected immediately on the phone so I clicked on <Sync> in the Contacts app at which point the change was visible. o be honest, I never really checked how long things took for something to become visible on the phone.
If nothing still works, would remove the account, uninstall the Outlook mobile app and then reinstall it, add the Outlook.com account again and go through the <Contacts> app to turn Sync on.