The long black slot at the bottom of the board in your second-to-last picture is definitely a PCIe x16 slot. PCI slots have the gap bridge (where the blade of the card is split) at the far end of the slot and don't have the latch at the back.
That's not a PSU that gives me a whole lot of confidence in being able to power up much of a graphics card without breaking a sweat, and it's certainly not packaged to handle much "sweat", if you know what I mean. Typically, a well-designed PSU is going to have much better ventilation.
I'm not sure what to recommend, because changing out the power supply is not always trivial. I had one case where I had to remove the whole motherboard to get the PSU in and out safely. Take a look at how simple it would be to swap out your PSU and base your decision on that. If it looks simple and straightforward, I'd swap it out before putting in a graphics card, because cheap PSU's can kill components on their way out.
If you're going to go for the PSU/GPU combo, I would seriously consider buying a pair that could serve as components of the new build if it's not too far off. If a GTX 650 Ti Boost would satisfy your needs in a new build, why not just go for that right now along with an appropriate PSU? Sure, the CPU/motherboard will then become your bottleneck, but maybe you'd be surprised. Also, if you're going to be able to get a well-designed case, motherboard, CPU and RAM in the next 6-12 months, the extra money spent on "too good" of a GPU will not have been "wasted".
If the PSU upgrade looks like a bit more of a challenge than you're game for, I'd be really tempted to go with an inexpensive discreet graphics card that runs on slot power, like this one:
SAPPHIRE 100323-2L Radeon HD 6570 1GB 128-bit DDR3 PCI Express 2.1 x16 HDCP Ready Video Card
It would definitely help in that more memory will be freed up for the rest of the system, and there's no doubt you would notice an improvement, with less risk to your investment (only $40 after MIR right now) should the PSU do something weird on you.