Will my circuit charge the battery and power Arduino?
No, the maximum voltage of the LM2576 is 45V. There are also a few other problems with the schematic above:
 - You need to have an isolation transformer on AC mains for safety purposes
 - AC mains can have spikes over 400V, from lightning or other devices. You need protection from these spikes.
 - AC mains should be fused, so in the event of a fault, it becomes disconnected.


EDIT

A circuit like this with a transformer would be better, instead of a 7805, put your DC to DC there. Use a step down transformer to get the voltage to the recommended voltage of the DC to DC converter. 


Source: https://www.elprocus.com/steps-to-convert-the-230v-ac-to-5v-dc/