with visa issue on arrival
What do you mean by this?
I will be in Kathmandu and needing to get to India as fast as I can by public transport to Chennai.
The reality is that flights are public transport, and a flight is going to be the fastest option. Either way, you need to consider the total travel time from KTM to MAA - not just a 'fast' immigration checkpoint.
Do you think the Sunauli one would still be the fastest or will the Eastern border be a better option, the one near Siliguri (I believe there s only one on the East).
I expect you mean the border at Kakarbhitta / Panitanki.
If you are going to travel by land to Chennai, then whichever border crossing you choose will not change in any material way how long it takes to get to Chennai.
https://erail.in/trains-between-stations/new-jalpaiguri-NJP/chennai-central-MAS?view=d
https://erail.in/trains-between-stations/varanasi-jn-BSB/chennai-central-MAS?view=d
https://erail.in/trains-between-stations/new-delhi-NDLS/chennai-central-MAS?view=d
If you really need to get to Chennai as fast as you can, then you should fly. KTM-DEL-MAA - seven hours, 150 EUR.