[bojan]

Success.
Finally today, I found some time to try detect the radiation from Sun at 1420MHz.
The proof is here: there are clearly two tracks visible (2dB difference), the top one was obtained with antenna pointed at Sun, the lower one is when antenna was pointed away from the Sun.
The tracks were obtained with Max Hold for 30 seconds, for both cases.
The noise floor of the pre-amplifier is still significantly lower than that of Spectrum analyser (there is no difference in noise level on the screen between preamp ON and OFF), so I can add more gain to the system (so far it is ~36dB).

With SDR (Terratec USB receiver) the signal is clearly visible:
The lowest part of the waterfall display is with Pre-amplifiers OFF, the middle part is Sun, and the top part is away from Sun.
The other screen shot from SDR window is simulated transit (antenna beam was moved across the Sun).

The system works!
This is only the beginning..