Can a VGA monitor be used with a DVI/HDMI graphics card?
As per the Wiki article on DVI:
A passive DVI-to-VGA adapter. This adapter will not work with a DVI-D output. It requires a DVI-I or DVI-A output to get the analog signal to a VGA input (even if the adapter looks like a DVI-D). A more expensive active adapter (or converter) is required to connect DVI-D to VGA.
As such you would need an active adapter in either case and it shouldn't really matter whenever you get an active DVI to VGA or active HDMI to VGA adapter. They're both available in a variety of qualities.
An advantage of using DisplayPort to VGA could possibly a simpler adapter as there supposedly is some form of backwards compatibility available. At one point I read that there was actual electrical compatibility as the adapter indicates to the card that a VGA display is attached but I'm not able to remember where I read it. Hence I might be remembering it wrong or it might simply be false.
Any VGA only monitor in this day and age is seriously obsolete - most older monitors would be at least DVI capable. Your video card reflects that and has no ports that can be "passively" converted into VGA.
Looking at your ports, that's a purely DVI-D port. It won't work. More accurately, the adaptor won't fit, and the analogue pins don't exist.
If you must stick with that monitor, you certainly will need a DP or HDMI to VGA adaptor.
If you look at the Wikipedia article for DVI, especially the pinout, you'll see that DVI also can carry analog signals (the variant called "DVI-A"). That's why DVI to VGA adapters work: They take the the analog signals from the DVI-A pins, and route them to the corresponding VGA pins.
So if your graphics card doesn't support analog pins on the DVI connector (which it doesn't, it's a double link DVI port according to the specs, and in your picture it's missing the analog pins if you compare with the pinout on Wikipedia), then there's no way to convert it to VGA directly with a cable adapter.
There are other, more expensive adapters which are basically a small graphics cards which read the DVI/HDMI signal, and then produces a completely different VGA signal from that.
So use on of those, buy a different graphics card or use your old graphics card in addition to the new one, or buy a new monitor.
DVI and HDMI are basically the same digital signal. The main differences are that DVI has the optional analog pins, and HDMI provides for a copy-protection scheme (HDCP) that's necessary for some consumer electronics, but not an issue between computer and monitor. HDMI to DVI cable converters work fine both directions.