Sadly, since Ubuntu is no longer considering 32-bit architecture something worth maintaining you may be better off using Ubuntu MATE as a learning platform for navigating the MATE DE, then move to Debian proper which still support 32-bit, or leaving the Debian userland using solutions like CentOS, Slackware, Gentoo etc.
I would not recommend trying to mix-and-match things unless you want to invest hours of time into making a hybrid Debian / Ubuntu installation work, but if you insist on using Ubuntu MATE you could punch this into the terminal and see if your 32-bit machine will boot;
In
x-terminal-emulator
:sudo add-architecture i386 sudo apt update
Once that is done, you should be able to install a 32-bit capable Linux kernel and boot into it. However you will need to install 32-bit libraries for existing software to function on a 32-bit system, and with the exclusion of multiarch-support
you may have to fetch that from Debian for the latest copy available from here.
Honestly this sounds like something I could have a play with when I have nothing better to do.