1990 honda crx si transmission trouble

loose wire or corroded connection at the starter or alternator, if not that it cab be ignition switch too

it depends if you a a d series or b or any other there's different series so u have to go according the type of engine you have

High milage may need trans fluid replaced. Not the clutch hydraulic fluid but the gear oil/fluid. Could be the 1st gear syncro is warn too.

I would suggest an ignition module. If it is a rough idle at startup I would check the throttle position sensor. You can check that by starting the car and unplugging the TPS. If the car runs differently then chances are it is working fine.

run scanner for trouble codes, some auto parts stores   will do this for free

No dipstick...you verify by havin car first on level ground then.Then when on stable ground remove the 17mm filler plug on upper passenger side of transmission.if fliud flows out of filler plug than level is fine.if does not flow out whenplug removed.RECOMMEND STRONGLY using a honda brand manual transmission fluid in this transmission..it does wonders for shifting or gearing issues.

Have you adjusted the clutch cable enough to allow it to disengage the clutch ? try that first. Is this the original clutch?

I would recommend flushing the tranny fluid. you could go with GM synchromesh fluid found at the GM dealer and it will make the shifting buttery smooth. I use this all the time. appx 2 quarts.

have you checked your automatic shift linkage?