I bought a home in a rich neighborhood because it is close to my work place, but I think other school districts would benefit more from my money. Is there any chance to choose where the school funding part of my property tax goes?
In the United States, schools tend to be funded by property taxes locally, occasionally with an added income tax. Those funds go to the schools' local taxing authority. That taxing authority may cover multiple schools. At minimum, it will almost certainly cover an elementary school and a high school, but it often covers multiple elementary schools and may cover multiple high schools. For example, in many cities, all public (government-run) schools are funded by the same taxing authority.
If you had children, you might, in some places, be able to choose the school they attended and thereby direct some money to that school or schools. But that wouldn't be related to the amount that you paid in any way. That amount may be more or less than the amount that you pay. It would have some relation to the amount that the child pays.
There is no way to redirect the funds that you pay. They always go to the local taxing authority. If you want to pay a different local taxing authority, then you need to buy your property in an area covered by that taxing authority. For example, you might prefer to live in the city rather than a suburb.
You can donate additional money above and beyond your taxes to an individual school. Contact the individual school for details. But this won't change the taxes that you owe.
Schools also get funding from the state and federal governments, mostly state. Given that you are subject to taxation by a particular state, you can't transfer those taxes either. Same thing with the US as a whole. It won't allow you to choose to pay your income taxes to a different country without changing your residence (and might not then).