It might be possible with a script, but in general spreadsheets don't "push" changes the way you describe. The tend to "pull" data from cells.
It's usually more efficient to keep like data together in one table rather than scattering it in multiple tables, and then extract summary "views" of that data as needed, either by using the built-in filtering capability, or by SUMIF, SUMIFS functions and the like.
If you're doing a budget, highly recommend you have a look at the Personal Budget template at File > New in your menu. It keeps the data in the 'Transactions' table on the second sheet and shows a summary of that data in the 'Summary by Category' table on the first sheet. It can easily be adapted to your specific needs.
If you need to shows multiple months you might have a look at an "expanded" version saved here (Dropbox download).