Puppy Linux is ok but not really that good for everyday end user experience, bit chinky and not friendly … yes I use it but only for limited jobs.
Lubuntu wold be closer to xp on older machine
But if possible upgrade the memory and go for the 32 bit version of mint 18 mate.
But network connections issue sounds like a network WiFi card mint mate may solve this or not, but cheap WiFi cards or usb key may give you bit more life
Either way think of it as a learning experience and finding more out about Linux and how things work, puppy would give you that no doubt