Yes, it needs to be upgraded. If it will fit, which it should, I'd highly recommend the Antec VP-450. You won't find another unit that is AS GOOD as that one at a similar price. Way below the cost of other quality units that are comparable.
The GTX 750 is intended for use as a step up on OEM machines, so it will almost surely fit. It is a dual slot card, meaning you still won't be able to use the adjacent PCI slot, but that's probably no issue. Either card should work fine.
Then it's going to be about 15mm longer than a mini ATX motherboard. Look at you case and make sure your hard drive cages are not directly behind because the card will slightly go past the motherboard which is 150mm square. This card is 165mm long
Everything you need to know about the cards that will fit in your case can be found here. And it DOES say you HAVE to use a low profile card. A full height card will NOT work in your case.
Apparently though there is misleading information then on the HP website. It indicates for both the desktop (Lays flat) and micro-tower models it needs a half height (Low profile) card. If you have measured and are sure it will fit, I'm sure you're right.
IF, that is the case, I'm unsure exactly what you still need to know. It seems you've already determined the length is ok and the height is ok, and as I said, so long as nothing is installed in the x1 slot next to it, then the width is ok and it should be fine then. That just leaves making sure the PSU is capable and has the correct connectors for the card.
Yes, it needs to be upgraded. If it will fit, which it should, I'd highly recommend the Antec VP-450. You won't find another unit that is AS GOOD as that one at a similar price. Way below the cost of other quality units that are comparable.