ATtiny13A and 433MHz Transmitter

I cant tell you what is wrong with your code but the sleep function you are using looks a lot more complicated then it needs to be, this is what i am using on a similar project:

#include <avr/interrupt.h>
#include <avr/sleep.h>
const int buttonPin = 0;


ISR(PCINT0_vect) {
}

#ifndef cbi
#define cbi(sfr, bit) (_SFR_BYTE(sfr) &= ~_BV(bit))
#endif
#ifndef sbi
#define sbi(sfr, bit) (_SFR_BYTE(sfr) |= _BV(bit))
#endif



int main(void) {
  init();
  {

    pinMode(buttonPin, INPUT_PULLUP);
    ADCSRA &= ~(1 << ADEN); //adc off
    sbi(GIMSK, PCIE); // Turn on Pin Change interrupt
    sbi(PCMSK, PCINT0); // Which pins are affected by the interrupt

    sei(); // Enable global interrupts
    set_sleep_mode(SLEEP_MODE_PWR_DOWN);


  }



  while (1) {
    if (!digitalRead(buttonPin)) //just to avoid sending on press and release(sends only on release)
    {
      //send stuff here


    }

    sleep_mode();
  }
}

Another thought, not code related INPUT_PULLUP can waste a lot of power over time mayby try a external pullup mayby 1M or at least 500k. and if the 433 transmitter are the same cheap ones I'm using, try driving the transmitter vcc from a pin on the attiny instead this also saves alot of power, just a thought
Best of luck man.