How important is it to upgrade ship weapons in Mass Effect 2?

Since I follow the XKCD theory of gaming, I'm just now playing through the Mass Effect trilogy.

I'm near the start of Mass Effect 2, and having recovered Garrus I now have the option to spend materials to upgrade the weapons systems on the Normandy 2. What sort of priority should I put on these upgrades? The existence of this option implies that there's some ship-to-ship combat in ME2, though there was none in ME1, but without any reference point I don't know if it's a good use of resources.

(Answer without spoilers if possible.)

Solution 1:

The three ship combat upgrades (armor, weapons, and shields) are only relevant in the final mission. However, they are essential if you want the best result in it.

When the Normandy travels through the Omega relay, it is attacked by a collector ship. For each of the three upgrades your are missing, you will lose a squad member:

- Missing the armour upgrade will cause Jack to be killed.
- Missing the shield upgrade will kill the first character on this list who is recruited and not one of the two members in your party: Kasumi, Legion, Tali, Thane, Garrus, Zaeed, Grunt, Samara
- Missing the weapons upgrade will kill the first possible character from the following: Thane, Garrus, Zaeed, Grunt, Jack, Samara

Losing a squad member here causes you to fail the No One Left Behind achievement.

Solution 2:

Near the start of the game, upgrading ship weapons is not that important.

It is only towards the end of the game that they are very important.

(The suicide mission)

You do go toe to toe against another ship as you rightly surmised.

In the last mission, the survival of your crew depends upon your ship upgrades.

Solution 3:

In addition to the mission mentioned by Studoku, the Normandy ship upgrades also indirectly affect the ending of Mass Effect 3.

For each installed upgrade, the Alliance Frigate Normandy SR-2 war asset is given a 25 point increase to its Military Strength value.