CAC-1173 is for HDMI output.
The Apple Studio Display only takes Thunderbolt or USB-C DisplayPort Alt Mode input.
The MacBook Pro (Retina, 13-inch, Early 2013) may be limited to 2560×1440 60Hz. Maybe it can do 4K30 from the Thunderbolt port or HDMI port?
https://apple.stackexchange.com/questions/129653/early-2013-macbook-pro-with-4k
A CAC-1332 may allow the USB functions of the display to work (brightness control, camera, audio) but for some reason Club-3d replaced it with the less functional CAC-1333.
The CAC-1336 also allows USB and it can allow up to 5K or 6K width (but your MBP would still be limited to 1440p60 or 4K30). https://insights.club-3d.com/thread/cac-1336-operation/
I think the best connection would be to use an Apple Thunderbolt 3 to Thunderbolt 2 Adapter with a Thunderbolt 2 cable. This way you get up to 10 Gbps USB from the USB-C ports of the Apple Studio Display instead of just USB 2.0 (480 Mb/s).
For full 5K60, you would need an eGPU. Check eGPU.io for examples/solutions.