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The only way to get at least one path between 1 and 81 without having a step of 6 or more is to have these two numbers in two opposite corners and let each step in the path be 5 (see blue path in picture), because 81-1=80 and 80/5=16, and 16 is the number of steps between two opposite corners. Any other way to put them will make the path shorter which will require at least one step of 6 or more.

Let's say we put 1 in the top left corner (as in the picture). In the next square besides it or under it, which is involved in the path, we must put 6. That means that the other neighbour can not be 6. If it is more than 6 then that step becomes 6 or more. If it is less than 6 then we have a remaining path down to the bottom right corner which has 15 steps. The "best" we can do is to put 5 there (see first yellow square in the picture). We then get 81-5=76 to cover in the remaining path. And 76/15=5 with rest 1. So there has to be at least one step of 6 or more somewhere on that path.

So we will always end up with at least one step of 6 or more.