Would time speed up near a large body of negative mass relative to observers in micro gravity? Time tends to slow down near objects with large amounts of positive mass relative to observers in micro gravity.  Considering that negative mass is the opposite of normal mass and would time tend to speed up near a body of negative mass relative to observers in micro gravity?
 A: There isn't a simple answer to your question because negative mass turns out to be rather strange when we try and use it in general relativity.
If we take the static black hole described by the Schwarzschild metric then the time dilation for an observer at a distance $r$ is given by:
$$ \frac{d\tau}{dt} = \sqrt{1 - \frac{2GM}{c^2r}} \tag{1} $$
and since $M \gt 0$ we get $d\tau < dt$ i.e. time is slower for the observer near the black hole. You might think we could just put a negative value for $M$ in the equation, and if we did that then we'd get $d\tau > dt$ so time would be faster near the (negative) black hole. Except that it turns out the Schwarzschild geometry with a negative mass is perturbatively unstable, so equation (1) isn't reliable.
This is the problem with your question. The temptation is just to answer that yes time does speed up near negative mass, but making this rigorous is harder than it seems. As it stands I think your question is basically meaningless as you would need to specify in a lot more detail exactly what physical systems you were considering. And of course many of us would take the position that negative mass doesn't exist, so the question is in any case no more than playing with a physically irrelevant bit of mathematics.
A: Recall in Newtonian mechanics when you could write $\vec F=m\vec a$ or you could write $F=\mathrm d\vec p/\mathrm d t$ and everything was fine.
But then when you studied relativity you found out $m\vec a\neq\mathrm d\vec p/\mathrm d t$ so you had to make a choice. They couldn't possibly both equal the force.
The same thing happens with negative mass. You have a bunch of formulas that used to be equivalent that now aren't equivalent. And different people can pick different ones as the ones they want to keep and have their own consequences for a negative mass.
For instance, does negative mass mean negative energy?  If energy is given by $$E=\sqrt{(mc^2)^2+(\vec p c)^2}$$ then it's still positive, even when mass is negative. And if momentum is given by $\vec p=\vec v E/c^2$ which is the only correct formula that works for zero mass, then a negative mass object with positive energy will still have momentum and velocity point in the same direction. It's actually a tiny bit silly if someone uses a formula that doesn't work for zero mass to claim they know what happens for negative mass.
So it might be that negative mass is rather boring, and that everything interesting happens when you have negative energy. And if you look at general relativity carefully, mass doesn't actually even appear in the Einstein Equation. Energy and momentum and stress and curvature appear.
So the answer is that when we deal with positive energy, lots of different formulas are equivalent that aren't equivalent when we deal with negative and positive mass. So without experiments or good theoretical reasons to pick some subset of these many formulas that are no longer equivalent, the words "negative mass" will mean different things to different people.
A: 
Would time speed up near a large body of negative mass relative to observers in micro gravity?

It would if there was any such thing as negative mass. 

Time tends to slow down near objects with large amounts of positive mass relative to observers in micro gravity. Considering that negative mass is the opposite of normal mass and would time tend to speed up near a body of negative mass relative to observers in micro gravity?

As above. As CuriousOne was saying, there's a big issue with the negative mass. As per Einstein's E=mc² paper, the mass of a body is a measure of its energy content. You can take energy out of a body and reduce its mass, because a radiating body loses mass. But when you get to zero via say electron-positron annihilation, that body isn't there any more. You can't take any more energy out of it, just as you can't shorten a pencil to less than 0cm. I know people wax lyrical about this sort of thing, particularly when it comes to things like wormholes and other speculative exotica, but I'm afraid it's what's called a "non-real solution". It's a negative carpet. If you have a square room with a floor area of 16m², some will suggest that you could buy a carpet measuring -4m by -4m. There is no such carpet, it's a non-real solution, and so is the negative mass. 
A: Considering negative mass would have opposite behavior to normal mass in every possible way. For example, normal mass causes negative potential energy, negative mass would cause positive potential energy. Negative mass would have negative inertia, means, it would not stay in state of rest, or uniform motion.. It becomes quite complicated. Hope the negative mass does not exist.
However, just for the heck of it, consider an observer near earth, experiencing all effects due to presence of mass of earth. For imagination purpose, you can consider as if there is a negative (but same magnitude) mass body exactly in opposite direction at exactly the same distance. Would the rate of time change? I would not think so. Because, it would still curve the space but in opposite way, and time would still be slowed down in equivalent manner due to curving of space. The geometry would be inverted, not the rate of flow of time - my guess.
