Hi:
I see several inconsistencies with your schematic. First of all, with the two diodes the way they are connected one of them will always be reverse biased at any time, and the capacitor will never charge. All you need is one diode in series with the capacitor (anode connected to top of transformer secondary, and cathode feeding the 1K resistor, which then feeds the +ve of the cap. The -ve of the cap connects to the bottom of the transformer winding).
Another thing is that I believe 1uH is quite low for the transformer windings. I am assuming you mean that these are the values of the primary and secondary inductances. With that kind of values, you will need frequencies well into the RF range in order to get any transformer action. Then, of course, the capacitor will be way too large, and the diodes will not rectify if they are ordinary Si rectifiers. You should have magnetizing inductances of a few mH's at a min.
Finally, can I suggest getting rid of the emitter follower stage, and driving the transformer directly from pin 3 of the 555 through a coupling capacitor (to remove all DC from the transformer primary). The 555 (do not use the CMOS variety though) is capable of sourcing and sinking current. With a 1:1 transformer, and 1K in the secondary, the 555 should be able to handle the current. That way you drive the transformer in a bipolar fashion. Otherwise, you would probably need a diode across the primary instead of letting the flux decay on its own. In any case, if you use an emitter follower with a 1K resistor as shown, there will be barely any voltage across the primary of the transformer. Also, you should aim for approx. 50% duty cycle in your 555 output (R2 should be much higher than R1 to approach 50%).
Are you winding your own transformer? or can you get the specs of the transformer you are using? That will determine the frequency of operation you should select.
Hope that helps,
Jem