Grounded outlet with load for ungrounded outlet
If you install a GFCI the load side of the GFCI can power 3 prong Ungrounded outlets just use one of the stickers in the box the GFCI comes with that says “GFCI protected no equipment ground” this is a legal way to update to 3 prong ungrounded outlets.

The home having 2 wire no ground points to pre 60’s era fwiw. 
It is possible that it had some level of requiring or to just prior to 3 wire requirements.

The current code allows you to run a ground wire (not a new cable as past code required) to any convenient location as long as the circuit comes from the same panel. So if you only have 1 main panel you can tie into another circuit’s ground to provide a true grounded outlet instead of GFCI only protection. Note a plug in outlet tester that has a GFCI test function won’t work on a no ground socket but the test button on the outlet will work.