That is a 4-core i5 and it should be OK with a 1050 Ti if you want to play at FHD at 60Hz. In more demanding games you might need to lower some graphic settings a bit.
Yes that's an H81 motherboard which have PCIE X16 gen 1 slot. According to Intel specs it should be a gen 2 but Asus website and manual says PCIE X16 gen 1.
The board has UEFI capable BIOS and it works with 10 series graphics cards but that gen 1 slot might limit the card performance a little (marginal). I would get a motherboard with at least a X16 gen 2 slot if it is possible.
One more thing, what is the brand and model of that PSU?
If you get a 1050 Ti that needs a 6-pin auxiliary PCIE power connector your PSU should be able to support that. Does the PSU have PCIE...