The obvious solution to lack of mana is Essence Reaver, the godly good item for Ashe and most adcs due to the massive stat boost it gives in addition to the mana restoration.
But I had a thought. "Essence Reaver is the meta solution to mana shortage for Ashe, and muramana is incredibly off-meta for Ashe and probably for good reason...but if I were to want to run it on her, what build would I go for?"
I found a reasonably recent thread documenting the item on her, where the OP went into BoRK and Rageblade (a classic I love) finishing with Runaan's; this playstyle is apparently very good for dueling but very bad for teamfighting and requires you to get ahead; falling behind is a death sentence using it.
The thread points out why pragmatically speaking, if playing seriously, this is a build which probably isn't too viable; it could be viable in specific circumstances, but the myriad of problems with it would keep it from being consistently competitive. However, it seems that even the critics of the idea admit, "yeah might not be optimal, but it seems hella fun to do".
An alternative proposed was manamune into Youmuu's Ghostblade, presumably going into some lethality items, similar to the video which prompted this idea of mine. Mathing it out, if Muramane is 2400 gold and boots are 900 and then two lethality items (Youmuu's + Duskblade) are 2900 each, that'd be 9100 for 4/6. Add in Runaan's, 2600, that's 11700. From there, cheapest options: PD for dueling; Statikk for waveclear; second cheapest, wit's end for attack speed and mr; guardian angel for revive; edge of night for defense; spirit visage for health/regen, mr, and cdr.
This doesn't seem the most optimal. You'd want rageblade and/or a lifesteal item (sanguine or BoRK, preferably BoRK) in there as well, but both of those are more expensive 3k items rather than cheaper 2k items.
I think, in the ideal world, it'd be Manamune + one and only one lethality item (Youmuu's for movespeed, Duskblade for raw damage) + Boots + Runaan's + Rageblade + BoRK, but that's a more expensive 15200.
It's probably not doable, but it's still a build I'd love to try out in the right environment.