Totally possible. 2 million Australian's came very close in "20 bloody 20"
About 80% of your requirements were met in 2020. 12 days of total isolation is a little shy of your requirements. A few subtle tweaks is all you need!
Remember this image? It is computer generated and isn't real, but it is sourced from real data overlays:

So as a "warm up" to my real answer - Sydney came pretty close to meeting your requirements in late 2019 / early 2020. As multiple fires across multiple states merged into mega-fires, and multiple mega-fires merged together around Sydney's perimeter, and the media started experimenting with the term "gigafire", Sydney was cut off from all highways and the interstate rail, the airport was operating at low capacity due to smoke cover. The Sky was red and ash was thick - it was opaque to satellite and aerial photography (at least without IR photography). A few very long winding routes were still possible into the city - via Newcastle or along the coast IIRC, and the internet and phone lines remained live. However this came pretty close.

However on the opposite side of the country, it was even worse. Perth was completely cut off by bushfires. Perth is 1.9 million people, 6400 square km, and every sealed road between Perth and all other state capital cities were closed. There were food and medicine shortages. Many empty shelves. It was the first city in 2020 to run out of toilet paper. Fresh fruit and veg was also scarce.
400 km of highways were closed, and a remote service station on the only sealed road into Perth needed helicopters to deliver supplies to the 250 truck drivers who were stranded there. With fires at the time also burning around Adelaide (my hometown - centre bottom), both ends of the road were shut due to fires. The temperature hit 49.8 celsius (121.64F).
The longest continuous period of complete isolation for the city was 12 straight days, however as the fire was controlled and escaped again the roads were re-opened and closed. 22 days in a month had total isolation. People were told to stay indoors and work from home. Outdoor work stopped. It was a warmup for Covid.

The airport didn't completely close, but there were long delays and cancellations as the city got blanketed in smoke and visibility went to 0. Planes got diverted or turned around but the wind subsided sometimes and allowed planes in.
So - for your scenario, once the fire starts, you only need to subtly change the wind to get your outcome (sans internet dieing):
- Stronger ash cloud over the airport. Keep the wind steady and no-ones flying in or out.
- Stronger ash cloud over the seaport so people can't work. If they can't breathe outside, they can't unload ships.
- Keep the fires burning up to the highway for a few more days. If crews weren't able to push the fires back on cooler, calm days, the roads wouldn't remain shut.
- Satellite and airborne photography suffer from anything in the air, thicker ash and it'd be pointless.
- Air drops couldn't be done in strong winds and zero visibility.
To cut Perth off from internet, you'd need to cut 5 undersea cables, this is quite an ask and unlikely to happen all at once. Cutting power will accomplish this. Ash clouds will ruin solar, strong winds can overwhelm wind generation (they drain power to apply brakes), and 0-visibility could affect fossil fuel power plant staffing, but cutting power in the Australian summer will kill hundreds of people from heatstroke however. A carrington event may take out all the ICs, and thus all internet and phone connections, and blow all fuses, but after resetting your circuit breaker your dumb Air Conditioner may still work. If a few dumb fossil fuel plants come back on line, or were shielded, you could have a society that is temporarily phone and internet free, but still air conditioned and has fridges to keep some food.
So to start it all, a Carrington event may be the answer. That caused sparks from telegraph lines. Sparks from modern power lines on 45 degree days with strong winds would start bushfires.
So the sequence of events is:
- Hot Australian summer. Typically "Catastrophic fire conditions" 45 degrees C. 40kmph winds.
- Official advice from government on these days is "Leave your homes early, before a fire starts, as once it starts, it's too late to leave."
- Carrington event hits. A geomagnetic storm hit Perth.
- Knocks out most ICs, killing the internet and phones.
- Power lines spark out in the bush, starting numerous fires.
- Strong winds spread the fires quickly, crossing the Nullarbor highway and the Indian-Pacific rail line, closing Perth to all rail traffic and all road traffic not on dirt paths.
- Fires spread into inaccessible terrain around Perth, like the Jan 2021 fires did.
- These fires can't be fought by ground crews as they're too remote. We can only fight them from the air.
- Ash cloud blankets the city.
- Stay-at-home orders issued. It's too smoky to go outside.
- Planes are unable to land as visibility is 0. They all Go-Around and then end up diverting to their alternate (Adelaide).
- Ships are unable to be unloaded due to the Ash cloud.
- Airborne water-bombers are unable to fly as the smoke is too thick, meaning we can't fight the main fire in inaccessible terrain.
- As usually happens after these heatwaves, cycles of low pressure air come in, bringing cooler temperatures but stronger winds and lighting.
- these help spread the fire massive distances. ember-attack 20km in front of the fire is not unheard of.
How does it all end? - Floods or heavy storms
Typically Australian bushfires need a big soaking to go out. They can burn into Autumn in inaccessible scrublands being slowly attacked by water bombers waiting on a big downpour. These usually have their own issues like the Jan 2020 Canberra bushfire was weakened by a $500million hailstorm.
Once you've made them suffer enough, have one of the low pressure cycles bring heavy rains, and then a period of calm winds.
- Emergency services can get on top of this within a few days of this.
- Opening a road after a bushfire is quite involved. Safe-looking trees can reignite and drop flaming branches onto trafic, so the surrounding bush needs to be cleared if its burnt.
- All the signs, safety rails, and other infrastructure is burnt away, so speed is limited after a fire, further delaying resupply.
Multiple times they've been extinguished by devastating floods or cyclones, so scenes like this road covered to a depth of 1m while still displaying bushfire warnings happen quite often:
