I'm sorry Patch has had this issue and know you will be worried for him.
When it comes to coughing in a cat, it is something we take seriously and it is not especially common.
There can be several causes, some of which are more serious than others.
Given his age and signs, I'd consider:
- Fluid on the lungs
- Heart disease
- A lung infection
- A foreign body such as a grass blade
Given his age and the timeline, as well as given how he was breathing in that video, I'd actually advise a consult as soon as you can.
If he cannot be examined conscious, we may need to provide a mild sedation.
Chest x-rays could be considered here, so we can get a diagnosis. Almost all vets have direct access to x-rays nowadays.
His treatment will depend on what is going on and may include e.g. anti inflammatories, diuretics or antibiotics.
If struggling with transport, I'd consider asking a friend or neighbour. You also have the option of using a pet taxi or a mobile vet service.