TeamPulse is designed to run equally well with TFS 2010 and 2012. There should be nothing required to be done from the side of the TeamPulse application for the upgrade. If the TFS server details used by TeamPulse (server name, collection, path, port number and user credentials) and the process templates of the currently synced projects remain unchanged - than the synchronization should continue working without issues after the upgrade. (Since you are using TeamPulse R1 2013 - than you should already have Team Explorer 2012 because it is required even for syncing with TFS 2010.)
It is just advisable to make sure there are no currently running project synchronizations from TeamPulse (to do that from the UI you can just open the TFS Sync Information screen in the old Silverlight interface - for the project(s) that you are currently using.) Then stop the TeamPulse integration service for the time while the TFS server is not available. After the TFS upgrade - just start again the TeamPulse integration service. You can run a sync in order to make sure everything is functioning ok.
Let us know if any issue occurs.