Does neurotin work right away to help with sciatica nerve pain? Or is it something that takes time to build up in your system?

Time/dosage. Finding the right dose takes time. The drug has limited efficacy for nerve pain. When it works its good. Often it does not help. Worth trying.
Yes and Yes. It can work immediately or the dose may need to be adjusted over several days to a few weeks to get to an effective level . Sometimes , using a combination of several different classes of medications may be more beneficial such as adding a NSAID as well as a serotonin uptake inhibitor like Cymbalta (duloxetine) if no contraindications to any of these meds along with other treatment like pt & esi's.
Needs to build up. Most people will need to take somewhere between 600mg 3x/day to 900mg 3x/day to get relief with Neurontin (gabapentin).