How to archive a transactional table on another server?
Caution: 

Since you will be using SQL Server 2012 express edition (and sql 2008R2), it has a limit of 10GB per database size (for previous versions, it is still 4GB). So if your table is not going to grow that big with all the archive data for many years, you are good uptill it hits 10GB limit.

My first suggestion would be to use Transactional replication. It wont have that much of overhead as you are just replicating 1 table. If you set the agent to run continuously or every 5 mins, then you will have up-to-date data on the destination (subscriber). You just have to remember to not replicate delete statements.

Secondly, you can use SSIS and schedule it using SQL Agent Job to incrementally load the data into the destination server.