To put a final end to this situation, I would get a survey done on my property which would clearly outline the boundaries and get quotes on repairing that part of the fence.

If you get a loan to buy it initially, the entire platted lot will be taken as collateral, you'd have to survey the excess land off and refinance the duplex on a smaller lot to free it of any encumbrance, or survey off the excess and only finance the purchase on a smaller lot if the seller can do the subdivision for a new legal description.

Would a land survey cover this?  

Be prepared for seller to stall, send outdated, inaccurate data and goose-chasing. my due diligence shortlist includes but not limited to- seller's Schedule E of 1040, copies of operating income statements for previous 2 years, rent roll, lease agreements, insurance binders, utility invoices, vendor service contracts, warranties, old inspections, survey.

They may or may not have an exiting survey you can use. 

I think Brian posted a survey of a little over 1,000,000 apartment units and expenses came to 49.98%.

LOL dude, did you get an elveation survey from the surveyor?

TAR just released their survey of rent performance in Texas' big five metros.

WASHINGTON – March 9, 2017 – Consumer confidence in the housing market has hit an all-time high, according to Fannie Mae's Home Purchase Sentiment Index (HPSI) in February, rising 5.6 percentage points to 88.3 – a record reading.Five of the six components measured by the survey hit record highs as well, including the share of Americans who say it's a good time to buy a home, and those who are feeling more secure about their job."

I have the architectural site drawings, and they have good references for where the border is, but in theory, the neighbor could dispute any painted boundary...which would probably cause me to need a site survey.