You should be fine with the processor in that build but I would recommend upgrading your RAM to at least 8GB if you are going to be doing any medium to heavy gaming. This thread linked below also can help with your question.
Do you know the motherboard model? As long as you have an one open PCIe x16 slot I imagine you'll be okay. The 1050Ti doesn't need any external power from the PSU.
I'm hesitant to say yes only because it is an older board and it looks obselete so the software is iffy - the hardware will fit but you will be very close to the rear I/O.