Yes, it's fine. In an informal context, "No, sorry, I'm a bit skint right now" is a completely natural response. Someone might say it to a friend, but not to their boss or bank manager. I'm not sure I agree with Andy that it's 'northern' . I think 'skint' is used throughout the UK. Is it a British word? Yes, if your definition of British = Not something that an American or Canadian would say. It's also used, I believe, in Ireland and Australia/NZ, so the range is a bit wider than just Britain.

I think that would be an appropriate and informal way to say it in Britain. (hear from someone who lives there?) I knew what it means but don't think I have heard it said in the USA. Here we'd be more likely to say something like "I'm kind of/a bit broke right now."

I have never heard of the word "skint." You would say, "Sorry, I'm a bit short right now."

Your answer is 100% perfect, it’s slightly more a northern word but the whole of Britain will recognize it.

If you are talking to a British person this would be easily understood. I lived in England for a number of years and often heard this expression. However, in Canada, it is more common for people to say, "Sorry. I'm a bit low on funds this weekend" or "Sorry. I can't afford it this weekend." Hope this gives you a few alternatives.