NYC yellow taxis aren't required to take passengers to NJ (except to EWR) and metered rates don't apply, so you'd have to negotiate the trip and a flat fare with the cab driver each time you hail a cab in Manhattan. NYC taxis also aren't allowed to pick up in Union City so you'd have to call a car service or local cab company to bring you back (I doubt if you can hail a cab on the street in Union City).

You might talk to the client and see if they can recommend a Union City (or nearby) based cab company or car service that you can arrange ahead of time to take care of rides both ways (I believe NJ cabs can pick up in NYC if they're dispatched and taking you to NJ). But even then I'm not sure if you want to be going through the Lincoln Tunnel at rush hour every day.

You might want to look into the Port Imperial ferry. It'll be a more pleasant and more reliable way to get across the river, and from there you can either have a cab/car service pick you up or (depending on where exactly you're going) take the Hudson-Bergen light rail the rest of the way.