So you bought an SMD part without the ability to make a PCB for it. There’s not much you can do there. You can try designing a breakout board and putting it though BatchPCB.com. If you are good at soldering you can solder wires to those tiny contacts.
Sparkfun indicated there’s a tutorial for the touchscreen, not for the connector.
I just got two touchscreens a couple days ago and no connectors. One of the touchscreens I’m going to just solder small wires directly to the flex cable. I have a TSC2046 touchscreen controller and a breakout board designed for it, I just need to etch it. I can give my results after I’m done, but that wont be for a week or two and it’s not for an Arduino. I’ll order some connectors next time I order from Sparkfun for actual projects, rather than this test setup.
As for the flex cable coming out of the connector; I haven’t seen this exact connector, but these ZIF sockets usually have a locking tab. Judging from the picture the dark plastic part should be able to move (probably slide in and out) and squeeze the flex cable to hold it in the connector.