Considering that CR is just a guideline, it could result in a Deadly encounter for your party or an easy one as well. Their chances of succeed depends on several other conditions not considered by the CR calculation or definition, such as terrain obstacles, previous encounters, the current party resources at the final battle, active magic effects, and so on.
So we need another indicator.
Taking a look at the "Creating a Monster" section in the DMG, there are some guidelines there for determining the CR of a custom monster. Specifically on page 274, they proposed a Monster Statistics by Challenge Rating table. It displays several indicators for determining CR, one of which is Damage per Round.
This considers the monster's damage output per round, which is the average result of the damage rolls the monster can get every round.
So we can borrow this idea and make some calculations on what could the Archmage's average damage output per round in order to get an idea on how deadly or easy would be for the party.
Doing some math, here is the average and maximum damage for each of the most damaging offensive spells the Archmage has (provided in the link) considering higher level cast (level 9, if possible):
- Cone of Cold: Average is 54 and maximum is 96 cold damage per affected party member, if unsuccessful CON saving throw (average of a d8 is 4.5, times 12)
- Lightning Bolt: Average is 49 and maximum is 84 lightning damage per affected party member , if unsuccessful DEX saving throw (average of a d6 is 3.5, times 14)
- Magic Missile: Average is 39 and maximum is 55 force damage (average of 1d4+1 is 3.5 per dart, times 11 darts, rounded up)
The rest of the spells are mostly defensive or non-combat ones.
Moreover, you could also calculate the average damage the players can inflict and the average amount of hit points they can regain, per round, to compare.
With this in mind, you can get a pretty good idea on how your party will fare in this battle and adjust accordingly.
If you think your players can comfortably resist the calculated damage, then they will most likely fare well, even find the encounter easy.
The downside, it can take a lot of time to calculate everything, it's almost a simulation, but it's worth it if you want your party to have a challenging encounter.
Finally, as per my personal experience, I've done these calculations and the necessary adjustments before with a boss on a final battle . As result, the party had a hard time, many characters went unconscious during the encounter, but they managed to beat the boss after many rounds of intense combat.
In this case I also considered the party's and the boss and its minions' strengths and weaknesses, including on who to target specific spells knowing the corresponding attributes needed for the spell saving throws. I hope it helps you.