Well, this mesh uses segmented weight painting, so what I'd so is edit the weight painting of the rigged character in blender, possibly using one of the free mixamo rig tools blender has for directly parenting a mesh to the mixamo rig and weight painting it, so you can just automatically use any mixamo animations with it.
I believe there are also data transfer tools that you could use to just transfer the animations to it directly after making the rig match the proportions of the character.
Basically you'll just want to select each part of the mesh and use the weight assignment menu and set each part of the mesh to the limb/head/chest bone of the rig as appropriate with 100% weight parenting.
The only exception here should be the fingers if I recall correctly, those you'll want more of a 50/50 weight blend around the joints.
Mixamo automatic weight painting does not tend to work very with well some lowpoly models, and esp not ps1 style one usually.
Sorry about the long reply time! c: