Last season, I lost a HOF caliber pitcher to a season-ending injury. He is slated to return four days into the season, or one day after the budgets are locked in. The injury is listed as "Elbow, surgery"

My medical budget hovers around 13-14. My question: should I increase that budget in the hopes that it will improve the effects of his recovery?

Here's the player in question:
 Hardball Dynasty – Fantasy Baseball Sim Games - Player Profile: Footsie Sanders

Normally, I'd say "yes" but you usually get the "Injury recovery" cycle one day before the player returns to full health.    I'm not sure of the processing order for this sort of thing.

Assuming the timing works out, should I up the medical budget as high as it can go?

Yes.   If it were me, and I wasn't already at 20m, I'd go as high as possible.   And I'd leave him on the DL until ST was over if I didn't get a recovery cycle during ST.

Thanks, mike.

Get it to 20 and never ever ever reduce it or training. 

I typically go 20M on training, and 16M on medical. I am not comfortable going below 16 on med.