Your intuition is correct. “We are going to go to the airport tomorrow by bus" is fine, it just emphasizes the mode of transportation. Note "by bus" and not "w

Your intuition is correct. “We are going to go to the airport tomorrow by bus" is fine, it just emphasizes the mode of transportation. Note "by bus" and not "with the bus", which is technically correct ("How are you getting to the airport?" "We're going with the bus.") but sounds a bit odd; "We're taking the bus" or "We're going by bus" would be more natural. After considering it more, I realized the only case in which "with the bus" is acceptable is when you are referring to "the bus" as a group of people, where you could plausibly say "the bus group". And that is why I considered changing the example to "with the shuttle" (as that sounded more plausible to me): because you are more likely to know a group of people on a shuttle than an entire bus.