Can I use the sentence "have you got an umbrella" or anything else to ask someone whether he got it or not?
"Do you have an umbrella?" can mean "do you own an umbrella?" or "do you have an umbrella with you right now?"
"Did you get an umbrella?" means "did you buy an umbrella?" (or get one as a gift or borrow one or steal one or however you might come to have one). That is, it is asking about the act of acquiring an umbrella, rather than simply having one.
If you asked someone, "Did you get an umbrella?", and he's had the same umbrella for 20 years, he might well say, "No, I already have one." Because he didn't get another, he's still using the one he already had. "Did you get ..." implies "recently", though there's no set period of time. It depends on context.
I think you are asking whether this can be about whether the person has acquired an umbrella, or only about whether they currently have one.
I can only answer for British English - I think the answer would not be the same for American English, where they mostly prefer Do you have to Have you; and use gotten for the past participle of get.
Here, there is no reason in principle why Have you got an umbrella? should not be a question about whether you went to get one; but because it is so common in the meaning of possession, it is unlikely to be used in that way. Far more likely are Did you get an umbrella? or Have you been to get an umbrella?