How do I build a switch-on circuit triggered by small signals, which will supply voltage for another circuit?
Whatever circuit you use to monitor the input to determine when to switch on the rest of the circuits still needs continuous power to do the monitoring.  Depending on the source impedance of the input signal (unknown), its bandwidth (unknown), its signal-to-noise ratio (unknown), and whether monitoring only the positive half cycles is good enough (unknown), the monitor circuit could be as simple as a single CMOS opamp acting as a very low power comparator.  A potential problem is that most CMOS opamps have average to poor input errors, plus those errors drift within the operating temperature range (unknown).