Thai banks have distinguish between mobile banking and internet web site banking even though both services utilize the internet as the medium. I can no longer log in on my computer to do banking online. You can scan pay, inquire, withdraw, transfer, deposit, using mobile app.
You're non b visa is just an intent of your employer that they want to officially hired you. But you are not yet employed because you do not have work permit yet. Besides, you don't need a tourist visa. The immigration will give you 60-day visa exemption.