How can I (masters in mathematics) prepare for an admission for PhD in economics?
I'd consider mathematics as an excellent footing for economics, perhaps top graduate schools and their professors will agree, ask them.

I don't see that delaying your application to teach mathematics for two years will help, and I don't think that delaying to "[get] involved in a research project in economics with a local economist" or "working at a local research institute" will provide a significant advantage. But, top graduate schools and their professors may disagree. Again, ask them.

It seems you don't know whether executing your plan (or aspects of your plan) will help make your application more competitive. This makes your question rather personal (indeed, you've gotten at least one close vote - not mine - on this basis). I strongly recommend asking the institutes involved, the relevant professors, or a combination of both, they can guide you, the broader Internet cannot.