Suppose 62 is an exact number and 5.67 is an approximate number. If we find their product we have 351.54. Since I found their product, would this be given to 3 significant figures (352)? However, if I add 5.67 to itself 62 times I would give my result with the same precision as the least precise number which would give 351.54 as my final result.
Would someone be able to clarify the significant figures rules for finding the product of an exact and an approximate number?
Significant figures are an approximate way to indicate the precision of a number. If you assume that your $5.67$ is properly rounded, the correct value should be in the range $(5.665,5.675)$. Multiplying by an exact $62$ gives a range of $(351.23,351.85)$ with the center being $351.54$ as you found. Writing $352$ indicates an uncertainty of about $\pm 0.5$, which is about what you have.
When you add numbers and keep the digits until the least precise value you are implicitly assuming that the errors are randomly distributed and will cancel out. When you add many copies of the same number that is not a good assumption because all the errors will be the same. Even if you add uncorrelated numbers, you expect the error to build up as the square root of the number of things you add, so adding $62$ independent values of $5.67$ you should probably report $351.5$ as being the best reflection of an uncertainty of $\pm \sqrt{62}\cdot 0.005 \approx 0.04$.
If you want to be careful, you should determine the range of each value that goes into a calculation, then consider which end of the range gives the minimum and maximum result and report the possible range of the result.
