1) Your transistor as shown can only discharge the cap via the relay coil. There is no means of charging the cap.
2) The circuit requires connections to both a positive supply terminal and a negative terminal.
3) Another resistor is needed, from the transistor base to whatever drives the base.
Change the circuit so that (a) the emitter of the transistor goes only to the supply negative and (b) the collector of the transistor goes only to the bottom of the relay coil. The coil should then energise briefly when the transistor turns on.
If the latching relay is a single-coil type (as shown) you will have to reverse the coil connections to reverse the relay contact state.
If it's a dual-coil type you will need one transistor for the 'set' coil and another one for the 'reset' coil.