You could develop a custom WebPart or Add-in for SP 2013 and in c# code of this project You could, using SPOnline CSOM, add items to SP Online.
The main problem is the authentication model You are using in SP-OnPrem and SP-online (OF365 tenant). If You have the same kind of authorization provider (for example in both places You are using adfs) then there should not be much of a problem to (as the same user account) submit some data from OnPrem to Online with some Add-in or Webpart as the user will be authenticated in both places. Otherwise You can submit some data from OnPrem to Online with CSOM but You will need to login (in code.. like hardcoded) as some user. This is not the best option from security point of view.
Here is an example how to login to SP Online using CSOM (You could do something very similar in the webpart or Add-in).
... so this are only ideas but generally to do this kind of logic first You need to figure out how You authenticate between OnPrem and Online. If You authenticate the 'same way' You could just create the context of SPOnline and then add/delete or update items with the user permissions. Otherwise You need to authenticate to SPOnline as some user to submit data.