reciprocity of antenna

If an antenna is used for the transmission purpose at some frequency band, is this antenna is good for reception purpose on same frequency band?

it can be used.. example is RADAR !

In general yes it is.
The antenna has the same properties (impedance, radiation pattern, gain) in the transmitting and the receiving case.

Of cause. It is generally reciprocal except that it is designed to be nonreciprocal (only some special applications).

If it is a "standard type" of antenna, it will work identically while transmitting or receiving.

If it was some new fangled fractal shape with metamaterials lenses, etc, etc, it may or may not work the same in both directions...depends.