I will be traveling to Chiangrai from Singapore by Thai Airways for 30 days. There is no direct flight from Singapore to Chiangrai but a stop over at BKK is required. Does the time that I transit added to my stay? As the transit flight is past midnight, it is on the 31st day. 

Yes, your transit in BKK is added to your stay. The start date of your stay is the day that you receive the entry stamp in BKK. 

That means I have to pay USD 100 to change my ticke????t. How long does it take to apply for Thai Pass?

42 minutes ago, CharTe said:

That means I have to pay USD 100 to change my ticke????t. How long does it take to apply for Thai Pass?

Nah ...change nothing. Just explain  the problem, act dim and pay a moderate overstay fine..You don't need to change your ticket

9 hours ago, CharTe said:

That means I have to pay USD 100 to change my ticke????t. How long does it take to apply for Thai Pass?

There is no overstay fine for a one day overstay it leaving from either airport in Bangkok. 

You can also apply for a 30 day extension of your visa exempt entry for a fee of 1900 baht.

It only takes few days (48 hours is shown on the site) to receive the Thailand Pass after applying.

You may not have any issue as your "exit day" is the date when you pass through immigration, not the date that your flight departs.  I do this frequently in BKK with a flight that departs after midnight and my exit stamp is dated the day that I pass through immigration, even if it is 11:59pm when I receive the stamp.