The speaker is making fun of articles that provide ridiculous ideas for saving time. He says that their suggestions are often very silly. As an example, he takes the idea that saving half a minute off microwave times will contribute toward having an extra hour of free time over a day.
You are correct that getting in on the bottom side of that refers to the three minutes (the lower end of the cooking time range). The phrase getting in on the bottom is sometimes used in situations such as the stock market or the housing market. For example, in the USA, housing prices peaked in 2007, but then crashed. They were at their lowest in 2009, and have since risen again. So in 2009, some astute person might have said:
This is a good time to invest in real estate. If I buy property now, I will be getting in on the bottom of the market.
The more usual preposition, however, is at rather than on. But on the bottom is often used in a metaphorical way with rung, referring to a ladder. To get in on the bottom rung means to start a low-paid job in some company with the hope that you will be able to rise through the ranks.
A similar phrase is get in on the ground floor, which uses an elevator as a metaphor. As the Collins Dictionary says, it can mean the same thing as the bottom rung phrase. But it has an additional, separate meaning: being involved in the early stages of some enterprise, such as a startup, so that as the enterprise grows and succeeds, your fortunes also rise alongside.
In all these cases, the idea is that something that is small or insignificant in itself will lead to greater success later, or that something which is low now will be high later. By using this particular phrase, the speaker is humorously tying the lower cooking time to such "bottoms" as the stock market, the ground floor, the lowest rung, etc. He is being sarcastic about the idea that shaving half a minute off by using the lower end of the suggested range will lead to greater success in saving time overall: as though nuking something for three minutes rather than 3.5 is likely to lead to huge success in the same way that getting in on the ground floor, or on the lowest rung, or at the bottom of the market could.
To address your question about the phrasal verb: it is get in. The on introduces a prepositional phrase that functions adverbially.