You can use the same construction in French. But it requires an imperative.
Regarde ton téléphone dans le train rien qu’une minute, et tu ne manqueras pas de souffrir du mal des transports.
Like in English, “et” can be omitted but usually isn't (the construction is hard enough to parse as it is).
This doesn't work in your context. You can use the second person imperative either to refer to the person you're speaking to, or sometimes to refer to people in general. It doesn't work if you don't want the statement to apply to the person you're talking to. There's no imperative first person.
You can use a subjunctive to work around the lack of imperative. It's rather literary, and it doesn't sound natural at all with this sentence.
Que je regarde ton téléphone dans le train rien qu’une minute, et je ne manquerai pas de souffrir du mal des transports.
The way I naturally express what you wanted to say is
Il suffit que je regarde mon téléphone une minute, et je souffre du mal des transports. (or: … et je me sens mal dans les transports.)
This is everyday spoken French. A more formal way of saying it, which would be more natural in writing, is
Il suffit que je regarde mon téléphone une minute pour que je commence à souffrir du mal des transports.
In all of these constructions, the subject of the second clause doesn't have to be the same as the first.
Parle trop fort au téléphone dans le train, et les autres passagers vont te demander de te taire.
Il suffit que mes yeux soient fixés sur mon smartphone dans le train rien qu’une minute, et je ne manquerai pas de souffrir du mal des transports.
That last sentence isn't very natural, though. In French, when referring to a
body part, it's common to say “j'ai le/la … ” (or “tu as le/la …”, “il/elle a le/la …”, etc.) rather than making the body part the subject of the sentence.
Il suffit que j'aie les yeux fixés sur mon smartphone dans le train rien qu’une minute, et je ne manquerai pas de souffrir du mal des transports.