I must admit that the choice of the LDO is perfect. I assembled one of these board and can use it with any solar panel up to 12V

If you wish, you may substitute S-1313 ( http://datasheet.sii-ic.com/en/voltage_regulator/S1313_E.pdf) for the MAX8887EZK27 LDO

What is the max ampere capacity of 3.3V battery it can charge ? Is there any upper limit like 1000mAh or 3300mAh like that ?

Right you are. Live and learn (even at 81), and I even learned a new word -'vias.' Thanks.