@Trevor McCormick to piggyback off what others have said, determine if you are trying to maximize it as an investment or utilize it as a vacation home you will STR to help cover expenses, as this could dictate which type of property you are looking for. 

Where / How can I learn what neighborhoods in Syracuse and the surrounding neighborhoods would fall into those (subjective) categories?

Once I get a few stable long term rentals maybe consider looking into such properties and cover aspects I am able to confirm. 

I’ll cover a few of those for you as well as try to offer some advice.

They cover the entire renovation expense.  

We recommend you get management contracts from several PMCs and compare the services they cover and, more importantly, what they each DO NOT cover.

For best results shop the vendors (who can be covered by insurance/meet HOA requirements) and get several options to your HOA management company to have the board pick from.

I'd look at the closing docs for something related to the pro-ration of expenses etc and see if you have anything there that could be a clause you point to for them to cover those payments. 

You may find this a viable option with rates where they are, especially if your rental income continues to cover the mortgage.Home Equity Line of Credit (HELOC): If your property qualifies, you could use a HELOC to access the equity without disturbing the current mortgage.

Hopefully the deposit covers Jan rent, make sure you give them an itemized list of the money owed to you for any damages or unpaid rent.