I’ve never heard is such an expression. To string-up something is like hanging it, which is very violent; so I wouldn’t use this in this context. I think maybe you mean to say ‘I am on a tight budget/ I can’t spend much money’: so in which case you could say ‘I need to keep a hold on my purse strings’. This would mean you don’t open your purse (old-fashioned purses would be opened and closed using a loop of string (like the hood on a hoodie, so if you keep hold of that string it means you don’t loosen it to open the purse).

I've never heard that. Maybe it's an expression somewhere in the English speaking world, but not in the US. I think people here would be confused. Also, "string up" generally means to hang something from a string.

No