Estimation is part of the solution for problems involving the floor function.  It just need to be some method that can estimate the radicals without the need of a calculator.  In part (a), if you realize $44^2 < 2020 < 45^2$, then it is good.

For Part (b), intuitively $\sqrt{800^2+1}$ is a tiny bit bigger than $800$, and $1-\sqrt{2}\approx -0.4$, so the "tiny bit" cannot compensate for $-0.4$, and the answer should be $799$.  Still, we want to get this result rigorously.  So we compare $\sqrt{800^2+1}-800$ and $\sqrt{2}-1$, and see if the former is indeed smaller than the latter.  This can be done by rationalizing:

$$\sqrt{800^2+1}-800 = \frac{1}{\sqrt{800^2+1}+800}$$

and

$$\sqrt{2}-1 = \frac{1}{\sqrt{2}+1}$$

Clearly the former denominator is way bigger than the latter denominator, therefore the conclusion is proved.