The first thing you need to understand is that the two pulleys on that shaft are secured to it using half-moon (Woodruff) keys and those must be removed prior to driving out the actual spindle. That sentence was put there for other members who may read this thread. As I understand it, you already have the upper and lower pulleys removed from the shaft. Just pull out the Woodruff keys and slide the shaft out of the bearings. If the shaft does not slide easily, then put the nut back onto it until it is flush with the top of the spindle and use a 2 LB hammer to strike a short piece of 2 x 4 laying on the nut. There's a chance that the lower bearing will come out with the shaft along with the bearing spacer.
If the lower bearing stays in place, then you need to to use a long punch to knock the bottom bearing out. The bearing spacer can be pushed slightly to one side so that the punch can strike the inner race of the bearing. Depending on how rusty everything is, the bearing may take several strong hammer blows to dislodge. Once you have on bearing out, the other one is easy. Just flip the deck over and punch it out. The snap rings stay in place. There is no reason to remove them unless they have been damaged in some way.
This deck has sent you a message. REBUILD ME. When one bearing fails, the other five are not far behind. Order six new bearings from one of our resident dealers and put this problem behind you. Other areas of concern are the two gauge wheels that support the rear of the deck when mowing as well as the belt idler pulley, the idler pulley tensioning spring and the idler pulley bracket PIVOT. Look at the belt for cracking and fraying. The replacement cost of this deck is more than $1500.00. Spend the money to replace the worn out parts now and enjoy at least four years of trouble-free grass cutting. Check the shell over for areas of heavy rust, pin holes, broken welds, cracked steel, bent parts. You have lots of time right now to conduct a thorough refurb. Our Mod Brad (Case220) completely redid his deck and has a thread showing what he did.