Well, That army you posted has a great weakness... No cavalry. Place your pike phalanxes (or like I said buy some cheaper ones like deuteroi or Machimoi) on the center just to bid your time (your elite phalanxes are too expensive to field enough. He'll keep outflancking you) and place your more elite infantry on the flanks...
Your cavalry should start behind your heavy infantry.

Well, like this:


PD - phalangitai deuteroi
EA - elite africans
AI - Assault Iberians
BS - Balearic slingers
SB - sacred band
HC - heavy cavalry (either sacred band or Iberi Lancers)
SC - skirmisher cavalry (Numidian or Iberian you see what fits you better... More missiles or the ability to better engage infantry.)

-------BS BS SB PD PD PD PD SB BS BS
----EA AI EA-------------------- EA AI EA
HC SC---------------------------------HC SC

Now, your Slingers will have to cope with his front row of slingers... while your heavy cavalry charges them from the flanks (if they're in wide good for you) you must engage and disengage all the time and don't forget to Alt charge!
Your sacred band and phalanx are gonna be missile absorvers... maybe move them first so he focus his missile units on them...

Don't haste yourself into battle take your time. Your heavy infantry has javelins his doesn't. And your infantry is more than able to cope with his Spartans and the ekdromoi Hoplitai no problem... Your center will have to hold a just the time needed for your superior heavy infantry to rip trough his.

If you don't know how to use slingers, get some Syrian Archers to do their work... just remember to get them to the flanks...

Inovate, try new things :) with this army composition you have the troops to take him. Once your engaged use your numidian light cavalry or the Iberian one to outflank them. unload missiles on the light hoplites backs to see them crumble faster than you can say "Bullseye!".

Cheers...