The Mini requires an input voltage of 7-9 volts so a single 3V battery will not power it. I have no idea what 433mhz transmitter you are talking about, so I can't comment as to what it's power requirements are.
I tried to connect the 3v arduino mini, it works without a transmitter (tested on basic blink sketch). But when I plug the transmitter then the signal from it is almost zero. This low voltage is safe for arduino?
The problem you are going to run into is current capability of the battery. That transmitter probably draws far more current than the battery is capable of providing.
How much current does your Arduino + Transmitter draw? (You need to measure this.)
Are you using the 3v3 version of the Mini or the 5v?