I own a MacBook Pro 11,1 (Late 2013) with an i7 CPU (2,8 GHz) and 16 GB memory. A machine perfectly capable of running Big Sur. Why should it not run Monterey as well? Well, because Apple decided so. Fortunately there is OpenCore Legacy Patcher. Find the download, as well as very good instructions here: https://dortania.github.io/OpenCore-Legacy-Patcher/
There is only one hint I can give from my experience: After booting into OpenCore and starting the Monterey install procedure, there comes a point, where the installer reboots (about when it says, that there are 12 minutes left). In my case, I ran into the „looping back into the beginning of the setup“ as described here: https://dortania.github.io/OpenCore-Legacy-Patcher/BOOT.html. Other than described here, I had to first select EFI Boot again (this maybe because I also have a Fedora Linux installation on this machine, that has its own EFI boot option in this menu) and then look for the „macOS Installer“ entry on the following screen. And, it didn’t have the circular Monterey signe in the back as shown in the OpenCore manual, it was just the hard disk icon. It took me a moment to figure that out, but now it works and I am writing this post from this very machine.
For my machine, the OpenCore website says that all functions work. What surprised me most: even Universal Control works flawless and its really great! SideCar does not, though. You can enable that with some tricks, but I reversed that due to bad performance and graphic glitches.
My thanks to Dortania!