Does the existing building need an injection of capital to complete rehab, and thus boost NOI? 

The money is deposited into your business account and on your balance sheet this would be considered a "cash injection" to the company. 

Then, when it's time for them to inject the capital, they develop a case of alligator arms and they struggle to reach their check book.

Many people find that they are already good at something and use that income to inject into the deals of others... 

The bottom line though is you'll likely need more equity capital injected into the project to make the deals in this day and age cash flow positively. 

How much cash have you injected into the deal at this point?

I've injected 10k cash into the house to update everything (the prior owner rented it out for a decade and had a lot of deferred maintenance and tenant damage).

All my prior deals with others have been 50-50 on money injected and liabilities assumed.

On the real estate investment side, we are using some shorter-term leverage and injecting cash into flips and shorter-term holds, but we're not maxing out what we can borrow.

Both of those options puts you in a better situation financially and leverage wise than total cash injection into one rental.Kind of like the saying "Do not keep all your eggs in one basket" is one way to think about the risk/reward.