If the input data is lat/lon referenced to WGS84, you could just use the Define Projection tool to change the definition to match what ArcGIS expects.
Because the output fc is empty, I'm wondering if the input coordinate system is defined incorrectly, so when the Project tool tries to reproject it, the coordinate values are corrupted. If you think that this is the problem, could you post the data's extent plus a general idea of what location it represents (in lat/lon or nearest city, county/state, etc.).
Thank you for the data link. That makes things a lot easier! ArcGIS can read the .prj file that contains the coordinate system information, and will correctly interpret the data as using decimal degrees in a geographic coordinate system. It should line up (overlay) correctly with your other data in WGS84 as-is.
However, the "WGS 84" names for the GEOGCS and DATUM don't match the Esri definition of "GCS_WGS_1984" and "D_WGS_1984" (although we will ignore the GCS_ and D_ when comparing names). Thus, you may be a warning message about different geographic coordinate systems/datums and that a transformation may be needed. You can ignore the warning too. If you plan to reproject the data into a different GCS or do certain types of processing, it may be easier to update the data's coordinate system definition to the ArcGIS one.
Using the Define Projection tool, select the data. For the coordinate system, browse to geographic coordinate systems, world and select WGS 1984. OK everything and you'll be good to go.