Yes that is correct. The tickets on the Green Bus to Mae Sai were 315 baht per person each way in January. This was the cost for comfortable armchair recliner like seats. It would be pretty impossinle to arrive in Chiang Mai on Monday and do your border run the same day. Unless you are arriving in Chiang Mai at 4.00 or 5.00am. You need to be at the Arcade Bus Station by 7.45am to catch the 8.00am bus. The bus station is not near the old town and you could find yourself in peak hour traffic that barely moves. It's risky trying to get out there straight after arriving in Chiang Mai. Have a look on the Green Bus Thailand website and see the seat availability. You can book your seats like you would do on an aircraft. You can pay for your tickets by credit card or at any
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. The 8.00 am bus for this Tuesday does not have 6 seats near the front available, people will have booked those well in advance. The 6.00am bus has plenty of seats, this is not such a comfortable bus but it is adequate. It will cost you 319 baht per person each way. Leaving at 8.00 will get you to Mae Sai about 12.30 depending on traffic and road works. It is not far to the border from the Mae Sai bus station but not walking distance either. Allow about 15 mins by songteow to get to the border. I would reccommend booking the 3.30pm bus back to Chiang Mai to give yourselves adequate time to get through Thai and Myanmar immigration both times, time to look around the quaint little market in Tacheleik and maybe grab a bite to eat. It's a long day without food and you don't get to stop for food on the bus trips. Bring snacks and plenty of water. Conversely though your idea of flying directly into Chiang Rai is not a bad idea at all. You would save money on the bus journey on the way up to the border and still book to travel to Chiang Mai on the Green Bus the next day or later. That way you could have a bit of a look around Chiang Rai and see the white and black temples.
Or you could do as others have said and go to immigration and do your extensions there for 1900 baht each. It is likely to be chaos at immigration this week as they will have just moved back (24th September) to their original building near the airport, about 500 metres from the airport. It is walkable, if you have a lot of luggage you might find it easier to leave it in the left luggage facility in Chiang Mai airport