It looks data for your city of interest (Elizabeth, NJ – code 2050) is available in the following years: 1960, 1930, 1920, 1910, 1900, 1880, 1870, and 1860. NATIVITY and BPL are also both available for those years, so yes, you should be able to obtain the information you are looking for.
The data you extract is called microdata, which means that is composed of individual records containing information collected on persons and households. In order to obtain the percentage of foreign born persons from Elizabeth, NJ, you will need to do so with a statistical package (i.e. Stata, SPSS, SAS). Keep in mind that for most samples the data are weighted. You can find individual sample descriptions here.
Additionally, you could look into using IPUMS NHGIS, which specifically provides spatially aggregated census data (data summarized within particular areas).