Generating a delayed pulse of 0,5sec with one 555 timer upon powering up the circuit
Assuming you're talking about akhmed's circuit:



The width of the output pulse is calculated in the normal way:

$$T_{ON} = 1.1\cdot R4 \cdot C4$$

The initial trigger delay uses the same formula:

$$T_{DELAY} = 1.1\cdot R1 \cdot C1$$

For both of these, you might use 470 k&Omega; and 1 &micro;F, giving a nominal value of 0.517 seconds for each.

The only "trick" required is that R3+C3 (via D2) must pull the trigger voltage above 1/3 VCC sometime after \$T_{DELAY}\$ but before \$T_{DELAY} + T_{ON}\$ (i.e., between 0.6 and 0.9 seconds). You need to keep the ratio of R3:R1 at around 1:3, and adjust C3 to get roughly the delay you need.

This is complicated by the forward drop of D2, but an approximation would be

$$T = -ln(1/3)\cdot R3 \cdot C3 = 0.4 \cdot R3 \cdot C3$$

If you set R3 to 150 k&Omega; and T to 0.75 s, you can solve for C3, giving 12.5 &micro;F. This isn't a standard value, but 12 &micro;F is, which should be close enough. Some experimentation may be required.