You've got two challenges here: automatically controlling the fan, and detecting a suitable trigger for an automation routine.

The very first thing I'd do, though, is see if the landlord can do anything about it before you do anything else. Free air flow between two units may be a fire code violation, or there may be a blockage further up the exhaust duct if the other fan is developing enough pressure to bypass your damper.

As for your question...

Controlling the fan is perhaps the easier of the two. A 4-speed fan switch most likely implements a muktivalue resistor with discrete values-- each ON position allows a different value of current to flow through the switch to the fan motor, thereby controlling the speed. There are wireless fan controllers out there, but they tend to be driven by RF or IR remotes, which CAN be incorporated into a home automation system, but it takes extra steps. I'm not really familiar with the options for operating those controls from a smart assistant, but others here might have better info than me.

Since you want the assistant to turn the fan on low, it might be possible to shim in a mini smart relay that can work with a rocker switch, like a Shelly 1 or similar, wired in series with the rocker switch. Then you can leave the fan on low while you are not using it, and allow the assistant to control it on demand. I don't know how those switches interact with a multi position switch though, since they are looking for transitions of ON/OFF to energize or deenergize the load terminals. You'd need to research it more. And, of course, you'd be modifying the range hood wiring which might cause you other, nontechnical issues.

As for detecting a trigger condition, this is rather harder. The changes in temperature, humidity or airflow outside your range hood when the neighboring fan is operating will be slight, perhaps within the noise band for the sensor, so my may get false positives, or just not detect it reliably. If you could situate the sensor in the duct itself, then airflow may be a viable option, but again, that may violate code or terms of your lease somehow.

Detecting odors is an altogether different set of challenges. Most of the common gas sensors are looking for specific compounds, like carbon monoxide, carbon dioxide, methane, hydrogen sulfide, etc. Food odors generally only contain trace amounts of those sorts of compounds. Your neighbors would need to consistently cook dishes that offgas one of these detectable compounts, which themselves would need to infiltrate into your air space in sufficient concentration to be reliably detected. I'm not an organic chemist and I am not saying it can't be done, but I wager a lot would have to be "just so" for your system to work as you describe it.

If it's not too intrusive for you to do so, you might just consider leaving the fan on low through mealtimes, perhaps using one of the micro relay suggestions in a timer routine on your assistant.