The answer is...........It Depends! It depends on what you intend to do with it. Do you need to use existing center holes on workpieces that need additional operations performed on them that must run closer than that to the previously machined features? The no, it won't be acceptable. But if you don't intend to remove the piece after machining and then put it back in the machine that close, then it will be fine. There are better centers, yes, but unless you need the accuracy, spend your money on something else.
Please take this the right way. If you're not sure that you need it, then it's likely that you don't. Most hobbyists don't work to tolerances stated on manufacturing drawings, especially just starting out in the hobby. If you had a print that you were machining to that indicated a close TIR requirement, that would dictate your needs and there would be no question.
And on the wearing in question. Bearings generally only get looser, so I''d really expect that center to be better than the spec at first, but probably run a little warmer than normal at high speed, then as things settled in, the 0.0008 would probably represent the best average of their product representation over it's lifespan. In short, at first it would be tight and closer, then loosen up, but not all at once to 0.0008, but eventually get there. The makers would probably consider that it's life expectancy. Of course, if it serves your purposes, nothing says you can't keep using it as long as it satisfies your needs.
If it's your first live center, and the price is right, you should be fine until you can measure and object to that small amount of runout. Depending on how deep you want to get into this hobby, that might be a month, a year....or you may see that it's fine for everything you want to do.
We can't always get what we want but I've found buying cheap usually costs me later.