That looks like an original carb, also looks like an air-vane governor.
Best to go to the
Briggs Manual page and enter the Model (90102), Type, and Trim numbers and pull up the parts and owners manuals to confirm the carb and linkage. You'll need to put an extra zero at the beginning of the Model number, so it'll look something like: 090102-0131-01 (that's a number I tried and it worked, but might not be your exact engine).
To put a kart throttle cable on there, you're going to need a place to mount the cable sheath, and a way to connect the cable to the governor linkage if you want to keep the governor.
If you're removing the governor, you can hook the throttle cable straight to the carb throttle.