Being a good physics question answer we'll ignore a few things, what the ball is made of, why it isn't tearing it's self apart and where we get the energy to spin it up seem like good choices since we're interested in what it looks like not how we build one.
The trick to understanding how this looks is to realise that there are two different frames of reference here.
Firstly the rotational measure, if you're stationary relative to the ball (to it's centre) it's spinning at a speed such that the outer edges are 0.995c, we want to be side on to the spin so the first visual effect of this is that the side spinning towards us is blue shifted, and the side spinning away is red shifted.
As 0.995c is pretty darn fast by any measurement I imagine that the shifts will be pretty extreme, well outside the visible spectrum, I don't have the maths to hand to work this out exactly, but the colour shift should be smooth across the surface most shifted at the side and less shifted towards the closest point to you (which wouldn't be shifted at all)
Now we have the ball spinning what happens if is thrown past me at 0.995c as well, and how come nothing ever moves faster than 1c?
The effect that comes to the rescue is time dilation, and the easiest way I've heard of to think about this is to measure time in terms of the speed of light, using a light clock.
A light clock is basically to mirrors a fixed distance apart with a single photon of light bouncing endlessly between them, time is measured as the time it takes the light to move from one to another and back again.
If you place a light clock on the top of the ball and measure it's rotational speed as the time it takes for one rotation as 1 when you set the ball moving imagine that the light clock is now moving perpendicular to the direction that the light is bouncing inside it.
From the perspective of the ball nothing changes, bounces at the same rate (the speed of light is constant regardless of your frame of reference)
From your point of view watching the ball move past you at 0.995c however things look different. Because light moves at a fixed speed and the mirrors are moving sideways the light now has further to travel. The calculations for this are fun to derive, but rather than reproducing them I'll link to a nice article Google turned up on the subject.
from which we learn:
which tells us that the time for the ball looks 9.9874922% the usual rate, so the ball is now looks to be spinning more slowly, the outside is now needs to move at around 0.099375547c rather than our original 0.995c
That's all special relativity tells us, but it still leaves us with a problem. $$0.995c + 0.0993c = 1.0943c$$ so the edges still seem to be moving faster than light, this is as far as I've got thinking it through with the time I've got, there are other things to consider, the fact that there's spacial dilation caused by the movement which will reduce the speed of the edges, as well as the fact that the edges are technically an accelerated frame not a static one (they're moving at 0.995c, but they're accelerating towards the centre to prevent them flying off) which you can dilation based on general relativity by pretending that they're being held in place by a gravitational field.
So how does this all look (since that's part of the question!)
Well it apparently still looks like a sphere:
so I'm going to guess it looks like a sphere with a blue shifted and a red shifted side, anything more specific than that and I think you'd have to be considering how quickly eyes can respond, what colours and lights are in play, etc.
Hope someone can work out the last mile and get all the velocities under c, it's a little beyond me at the moment.