Part (a)
The manager could break up the employees into two gender groups of male vs female. Each group is a stratum and collectively we call them strata. So stratum is the singular form while strata is the plural. So that's where "stratified" comes from.
Another way the employer could break things up is to have those in favor of expanding the existing break room would form one group, while the patio people would form another group. This might be the better method. Once people are separated like this, the manager would then pick an equal number of people from both strata. Let's say there are 100 employees and there are 50 people per stratum. The manager would then randomly select perhaps 5 people per stratum to form a sample size of 5+5 = 10 people.
The reason why we break people into groups like this is to ensure that every facet of the population is covered. In the first example, we are covering men vs women, and not leaving out any one single gender. In the second example, we have break room vs patio as the two facets of interest. Perhaps the manager could employ both of these methods to get the best snapshot of what his employees prefer.
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Part (b)
The manager could use the rule "every 10th employee that shows up to work will be selected as part of the sample". We can replace "10th" with any number you prefer. So it could be "every 3rd employee" instead. The key is to stay consistent to the rule at all times if you use systematic sampling.
While this method seems random, there could be flaws to it in that there could be a rare case that (nearly) every 10th person is in favor of the break room and that would leave out any patio people. Or perhaps every 10th person is a male and that leaves out females entirely. These seem rare but you need to account for them. This is why stratified sampling is preferred in my opinion. There's also the risk of the manager somehow manipulating the rule so that the manager gets the results they want, rather than the true result of what the population of employees want. This manipulation may be intentional or it may be accidental, which is considered bias. The general rule of statistics is to try to be as random as possible.