The two big things are a way to conceptualize your problem in a straightforward way (hard) and to learn to rapidly do adequate (not precise) arithmetic in your head (easy).
Wait, the math is easy? Sure, if all you are concerned with is the right, not correct answer. Back when people used slide rules this was common, but now when you do it it seems to weird some people out. You should develop this skill anyway because when you see an answer you should be able to tell at a glance if it's probably right or almost certainly wrong.
Why its simple: the first part is just to keep track of the order of magnitude.* People who used slide rules always had to to do this, and it's quick to pick up and pretty easy to do once you're used to it.
Second is just to know a few common fractions and be comfortable rounding intermediate results to convenient amounts (if you have "86" you might round it to 81 if you're dividing into thirds or ninths, or 88 if its by 11, or 80 or 90 if you care about 10x and would prefer your error to be a "too small" or "too big".
Third is to understand those error bars above, and, as you do when you work by floating point, avoid dealing with incommensurate numbers (this factor is so tiny I'll just ignore it).
When you get good at this you'll usually be within a few percent of the actual answer, which is usually enough to decide if it's worth actually calculating the answer or not.
Example: I remember a discussion a few years ago where we were trying to figure out if we could fit our product into a certain volume. As we discussed the parameters, I and a colleague simultaneously said "360 micrograms" (density was 1 so g = ml). The calculator welder beavered on and a few seconds later triumphantly said "357 mg". Great, 357 was more accurate than 360, but about right, but it was clear he was madly off in magnitude. He wanted to believe his calculator, and checked his work while the rest of us moved on.
* Unless you're a physicist in which case within a few orders of magnitude is probably OK, or a cosmologist in which case all you care about is 10^0