Caravanistan

I am Indian citizen working/living in Saudi Arab (there is no Mongolian embassy or consulate here) how can I get my tourist visa. I am planning to visit in second half of May or early June.

One more question.

Is there a way to go from Mongolia to Kyrgyzstan without getting visa for third country in between?

Have a look at https://caravanistan.com/visa/mongolia for the general application procedure.

You can fly from Mongolia to Kyrgyzstan, probably with a transit in China will be easiest.