For the 100m, top speed and acceleration are all-important. Speed = cadence x stride length. For cadence, try to minimise ground contact time by running forefoot (preferably with spikes if you are accustomed to them), and look on the Internet for correct technique guidance in regards to knee drive and arms etc. For stride length, it is important to work on motor unit recruitment by training the neural pathways that 'recruit' muscle. This can be done through plyometric training, and drills such as ladder sprinting. Strength training with heavy weights will also improve your power output.
Make sure to work on your block start technique, either with a coach or based off YouTube tutorials.
For the 200m, your loss of speed is a combination of two factors: H+ accumulation (think lactic acid), and depletion of muscular phosphocreatine (PC) stores. PC is an energy storage molecule found in small amounts in major muscle groups to enable massively energetic bursts for up to ~15 seconds. You can train your muscles to store more PC through repeated very high intensity efforts about 95% of maximum intensity and about 5-12 seconds duration. Make sure to give sufficient recovery time between repetitions with a ~1:5 work:rest ratio.
As for tolerance of H+, you can improve this by training your muscles to store chemicals that buffer such an increase in acidity. On a practical level, this is done by training sessions consisting of repetitions 15-45 seconds long at 85-95% maximum intensity, with a work:rest ratio of ~1:3.
Make sure to stretch and warm up and warm down before and after training sessions, and try to allow maximum recovery and adaptation by consuming snacks with decent levels of carbohydrates and protein within 30 minutes after a training session. Also keep hydrated and sleep well.
For any significant adaptation, you will want to aim to be training for at least 6 weeks prior to your events.
So a sample week might look like:
Mon - 5x80m run throughs (50 sec rests) at 95% intensity (develop motor unit recruitment, technique and PC stores)
Tues - active daily life (ADL) (eg cycling is great)
Wed - moderate gym session focusing on major leg muscle groups with heavyish weights
Thurs - ADL
Fri - 4 x 150m (60 second rests) at 90% intensity
Sat - ADL
Sun - ADL
Mon - working on technique with block starts
Etc etc
Note: I mentioned plyometric for the sake of completeness in a guide to sprinting, but they can be VERY dangerous. If your body isn't ready or your technique is off you can do serious damage. I would say that you should not attempt these unless you have a coach to supervise and say that you are ready. I for example did some plyometric jump squats in November 2016 and haven't been able to cycle since due to knee tendon issues. Seriously, do not even consider these without supervision.
Good luck in your athletics!