If you are playing through mission mode for other achievements you will undoubtedly get this along the way quite easily, but if you want to charge to it as fast as possible my favored method is to repeatedly do the Gundam Fight Semifinals friendship mission on Normal which are unlocked by reading the terminal message from Rain about the Gundam Fight Tournament. You just have to kill one ace who is right in front of you- takes less than 30seconds. I like to swap between suits because on normal you will obtain rank 3 parts at least half of the time and it's easy enough that you can pull it off with any suit you ahve a liscense for assuming your pilot isnt level 1 or something, no matter how badly their parts are, so you're assuredly beefing up your suits at the same time.
I guess this is also the best mission to farm pilot skills as well (I guess you could do the first gundam fight on easy and 1-shot them, but that's really overkill since most of the time spamming this mission is loading and managing your mech lab... not the dismantling of your opponent in 5-20sec) since you have a chance to gain skills them after each mission. This being the fastest mission, cycling through suits will get you many skills as well as assured parts for your suit, and build the mission count in a hurry.
I farmed this mission for a while when I had 40 personal missions done with haman in mission mode to get to the 50 missions completed (with one character) which upon beating unlocks musha gundam mk. II as a suit that can randomly show up and complete your mobile suit gallery achievement(and once you've unlocked this event on one character, the musha 2 can appear for any character, so the earlier you knock out the mission to make it appear, the quicker you can assemble it and head toward the liscense (kill 5000 total-read terminal message for reward: musha 1 can appear then "remember the past" 1-6 for musha 1 liscense mission to unlock. then 50 total missions with one specific character- complete the 50 total missions mission, then "remember the past" 7 and then that unlocks the musha 2 liscense mission). Once you complete the requirement for them to show up, they'll show up very frequently. I see them in about half the missions I do that arent collection/friendship (ie. devoid of enemy aces).