In English, we would generally use this in the past tense. because at the time it is happening i.e. the present, you still think the other person is coming, just running late. So we'd say 'I got stood up.'
"I get stood up" means this happens to me with some frequency. "Am being stood up," or "am stood up" or "am getting stood up" would apply to a singular ongoing situation. If an English speaker leaving the scene of the event uses the present tense, he DOES NOT say, "I get stood up." Granted, he is more likely to use the past tense.
Does this mean 'stood up' as in - go on a romantic date and the other person doesn't show up? Or as in - fall over and be righted?
The former, but this doesn't necessarily apply to romantic dates only. Say you have an appointment to hang out with a friend but they don't show up, then this could also be used.