Actually as Ashe vs Akali there are several things you can do to stop her from immediately shutting you down.
- Hexdrinker (plus a little luck)
- Mercury Treads (because I know some Ashes who just don't buy Treads)
- Banshee's Veil / Guardian Angel
You can't stop her as Ashe really can't fight melee regardless of who it is. She's strong when she hits from max range forcing the person to close the distance or run away, but she can't take a hit. You don't want to lose this aspect though so Hexdrinker doesn't totally diminish your damage, but it does extend your health by 300 HP which isn't bad, but it does require luck for the damage that activates it has to be magic.
Banshee's Veil is also strong to just take that first blow, but unfortunately Akali's main source of damage isn't her Ultimate. Doesn't really matter since you just want to slow down how fast she kills you. It provides solid survivability on Ashe especially since she's such an easy target if she gets caught. A stronger solution would be Guardian Angel, but it doesn't provide you the HP/Mana benefits of BV. Akali will actually have a much harder time of killing you because after you die, it's a pretty long wait for the rez and Akali isn't exactly tanky either so she'll be forced to switch targets or leave the fight if your team is focusing her properly.
Saving your Ult for her instead of initiating is an option, unfortunately it can't always be the case, but if it gets to the point where your team isn't saving you fast enough, you're going to have to compromise.
Also, as a word of advice: Don't run away from your team and out of the fight. You're not going to survive or get away. You're actually just making it worse for yourself because your team isn't going to chase Akali to save you when 4 other people are there hitting them. Stay close to your team.