Alumni Tax Expert @ErnieS0 says New York follows what's called the "convenience of the employer" rule meaning many telecommuters must pay NY income tax if their employer considers their "office" to be the company's NY location, even if you never set foot in the building.
See: State Tax remote worker NY company living in MN
In your case, 100% of your income will be taxed by New York — days you worked remotely and days you worked in New York City.
Other state tax credit
You will file a nonresident New York return and a resident Ohio return. Ohio (not NY) will give you a credit for tax paid to the other state since OH is your resident state.
TurboTax will automatically calculate the credit on your OH return if you first prepare the NY return.
See: How do I file a nonresident state return?
City tax
You are liable for OH city tax since your are working from your home in OH. Most likely, your employer will only withhold NY state tax and not Ohio state or city tax.
New York has a higher tax rate than Ohio, so the credit for NY tax paid on your OH return should cover the tax liability from your job — so you won't owe OH tax. However you will probably owe city tax. You might want to make quarterly tax payments so you don't owe a whole lot at tax time.
You do not have to pay New York City tax.
