I lived and worked in NJ in 2018, but received approximately $1,600 in unemployment from NY State in 2018, do I need to file state return in NY?

Your NY unemployment is NY-source income and thus taxable by NY.  As to whether you must file an NY tax return, see "Who Must File" on page 7 of this reference: https://www.tax.ny.gov/pdf/current_forms/it/it203i.pdf

Your NY unemployment is not taxed by NJ.  Unlike NY, NJ does not tax unemployment benefits.