A competitive back-up goalkeeper to keep Mignolet on his toes - I would prefer Alex McCarthy. He's youngish, English and showed a lot of potential before he went to QPR, that one performance against us was unbelievably good.
A first choice right back to replace Johnson - I would prefer Dani Alves, but I think without Champions League football we'll end up with someone like Montoya on loan or maybe Micah Richards if he wants a return to England.
A first choice holding/defensive midfielder. Milner is supposedly a done deal, which would fit the bill as I can see him progressing into a holding midfielder at this stage in his career, sitting deep like Gerrard did for the 13/14 season. He doesn't have the passing abilities to be a "quarterback" like Gerrard, but he has the legs and the engine to protect the defense and link up play.
A first choice striker to partner up top with Sturridge and Sterling (if he stays), à la Suárez. This is where we'd have to spend big. We tried replacing Suárez's goals with the combined input of Balotelli, Borini and Lambert - but that didn't work. Those three should be out the door (although I wouldn't mind Borini staying, as he's more versatile and we'll need depth if we're going to be in the Europa League again). Again, it's going to be tricky without Champions League football. I think Reus would be ideal, but he will never consider us without Champions League football - and with Real Madrid reportedly after him. Lacazette could be a possibility if we spent big (£30-£40M depending on who else is interested), but Lyon look like they're building something special so he may not want to leave his hometown club just yet. Alternatively, I think Lavezzi could bet got. He's third in the pecking order with Ibrahimovic and Cavani at PSG and is often played out of position on the wing, so I think he could be tempted away. I think this would be the most realistic acquisition to go for.
If Sterling leaves, we should just go straight out and get Walcott. Yes, it's a step-down as he's older and doesn't have the potential of Sterling. But, he's the obvious solution to that position, and I think given a run in the first team he could gain back the confidence and form that we saw from him in his early 20's. It's a bit of a risk given he has been a little injury prone over the last couple years - but so has everyone at Arsenal, there must be something in the water.