You could set the dome up such that it had automatic fire suppression.
If atmosphere inside the dome was slightly below exterior atmosphere the outside would help hold the dome down onto the ground.
If there is a hole in the dome the outside atmosphere will come in. There is no oxygen in this atmosphere and it will not itself support combustion.
A fire inside the dome will burn as long as there is something to burn and oxygen to burn it with. The oxygen itself will not burn. The fire will burn trees, houses etc like fires do.
When the methane comes in and hits the fire and oxygen it will burn. It will compete for any available oxygen with the burning trees and houses. It will not explode. It will burn slowly, like the Hindenburg.
The faster the outside comes in the faster this will happen. You could have the dome open up to facilitate that. It is like those fire suppression systems that dilute the air with nonoxygen containing gas.
When oxygen inside the dome is consumed down past a point the fires will go out. Dilution of oxygen nitrogen atmopshere by outside atmosphere will hasten this.
Of course for a dome breach there would be sirens etc. Every human habitation would have oxygen cylinders and masks. People would put them on and head for the tunnels.
Later you could fix the dome and pump the correct atmosphere back in. Your buildings etc should still be in decent shape.
I am getting those pesky downvotes for lack of hard science. Rather than delete this answer in a fit of pique I will lay out what I think is needed to answer with hard science.
Here is what is needed for the hard science: the exact composition of the atmosphere. If 85% methane and 15% CO2 this will pose more explosive risk than vice versa. One could use data from mine operations to figure explosion risk.
Hard science consideration point by point:
Can city engineers shunt the inner atmosphere out in time to prevent
a catastrophic collapse of the dome?
How to answer this with science? How could shunting out atmosphere protect a dome? If this is an inflatable dome like the Minneapolis Metrodome (higher pressure inside than out), shunting out the atmosphere will deflate it and it will collapse under the exterior atmospheric pressure. If, as I think more appropriate, the interior is at equal or slightly lower pressure than the exterior, shunting out the interior atmosphere (and reducing interior pressure) will just pull in the exterior atmosphere - through the dome holes caused by the attack if nowhere else. Again, no catastrophic collapse.
Can the people stop the fire from spreading and reseal the damaged
portion of the dome or will the inferno happen so fast there won't be
any time for it?
If the inside of the dome is earth normal atmosphere, a fire would be like any other fire... initially. Maybe the OP is thinking about the Apollo 1 fire which spread very fast inside the cockpit because of the high O2 atmosphere. The proposed dome has an earth normal atmosphere.
Incendiary bombs like those dropped on Tokyo in WW2 make fires that are hard to control, whether in a dome or in Tokyo. The two variables in the dome not present in Tokyo:
Firedamp is explosive at concentrations between 4% and 16%, with most
explosions occurring at around 10%. It caused much loss of life in
coal mines before the invention of the Geordie lamp and Davy lamp.
I am not able to guess what % methane is in the atmosphere. The atmosphere of Titan has 1.4% methane which would not be explosive mixed with earth type atmosphere. It looks like on this planet, based on the metabolism of the aliens as far as I can understand, methane is in a role similar to that of CO2 on earth. CO2 is less than 1% of earth's atmosphere; again non explosive.
I am going to therefore guess that the % methane in this scenario is less than 4% and so there is no risk of explosion due to methane coming in and hitting the fire. The mostly CO2 atmosphere coming is will act like a CO2 fire suppression system which are commercially available and work great.
•Would the whole city in that particular dome actually im/explode? If
the answer is yes, how long till it happens from the initial attack?
This has a lot to do with the nature of the initial attack. If the initial attack is a colossal bomb it will produce a big explosion as they are wont to do. If my guess above was wrong and there is a high % methane outside and a fair bit can mix with the interior O2 containing atmosphere and be ignited, that will cause an explosion of an energy dependent on the amount of methane mixed with the O2. Any other attack will not make an explosion.
- If the dome is like the Metrodome and is held up by pressure, the heat caused by the fire will increase pressure. Domes of this sort have valves to vent pressure and prevent rupture and these will protect the dome.
- If the dome is lower or equal pressure to the outside there will likewise be valves to maintain this.
- If there is a hole in the dome caused by the attack the pressure inside and outside will equilibrate thru the hole.
- If the defenders of the dome vent the atmosphere pressure will likewise equilibrate.
- If there is a low detonation velocity explosion from methane and oxygen (measured at 1512 m/s^2 here); half that of the slowest detonation velocity of any conventional explosive as listed here) the hole in the dome plus pressure relief vents might be able to take care of the overpressure.
•How much time do the citizens have in order to seal off their
houses/buildings or get into underground shelters?
Houses can automatically seal if there is a dome breach or pressure change or fire. The time the citizen have depend on how fast the atmosphere in the dome will turn into 1: the outside atmosphere or 2: an oxygen poor / carbon monoxide rich atmosphere because of the fire. That depends on the nature of the fire (incendiary bombs can kill by depleting regional O2) and damage to the dome / intentional venting of atmosphere.
Since this is fiction you can adjust scenario to be what serves the story - people trapped outside banging on the door as the houses seal up and the dome vents. Or people strapping on their O2 masks and coolly watching the fires go out as the atmosphere vents.