Not sure were you mean you want to paddle too from the Fort Niagara? There is current on the lower but not a whole lot. I have paddle from Youngstown Yacht Club (there is a public ramp and parking) up to Lewiston launch by Silo Restaurant. Lewiston charges 5 dollar per kayak to launch from there. Easier to paddle downstream from Lewiston to Youngstown launch. Or paddle to Fort Niagara. If you want to launch or land in the Fort Niagara park it again cost money to get into the park. A nice paddle to do if the lake is calm for you is to launch from Youngstown Yacht club paddle across to Canadian side go out into lake along Canadian shore and paddle down lake then back to Youngstown. If you want no part of the lake then paddle either from Lewiston down to Youngstown Yacht club or to Fort Niagara. Keep in mind you would have to some how get a car down at the end if you want to do that.
No shuttle, then paddle from Youngstown Yacht club up river to you either reach Lewiston launch/Silo Restaurant or you get tired then paddle back. If you stay along the shore line paddling upstream is not hard, paddling back go farther from shore to get small current to help you back.
If your crazy like me paddle from Youngstown up stream to Queenston/ Lewiston bridge. Extremely hard! Can be dangerous. Heavy current and rocks. Not for beginners or slow kayaks. Helmet recommended. Thats way upstream Lewiston Launch.
If you launch from Fort Niagara on lake side which can be done paddling into the mouth of river has more current right there along the Fort and mouth. Just to let you know. I have seen 10 foot rec boats do it but a little hard paddling for about a 100 yards. Easy on Canadian side of river mouth, no real current there.
Youngstown Yacht club is free to launch and park. If lower parking is full you can park up top in free parking across the street. I usually go from Youngstown yacht club launch. Free and easy to do.