The cylinder is called a starter, if you google for "halogen lamp starter" you'll find those. Essentially they charge up and then release a spark that ignites the gas inside the actual bulb. They are very inexpensive, so I'd start from there. Doesn't hurt to have a spare anyway.
If you unscrew it, the correct wattage should be written on the bottom between the prongs, just make sure you get one with the same rating so it works with your bulb. Feel free to post a picture of it and I can help search for one.
The prongs itself should be standardized, but the length of the starter may vary, so check if you can still close the lamp.