I was a physics major in college. We will make all the common magical assumptions about this scenario: the stick is strong enough, no gravity, no buckling, no orbiting, you are strong enough, etc.
It would take 2.5 years at the very least for the stick to reach the star. It is impossible for it to take only the amount of time it takes to move your hand because of the limits on the speed of light. When we move shorter, normal sticks around, we assume the tips of the stick move at the same time as the part of the stick we hold. However, this is not true. In fact, all solids are made of atoms that are linked to each other through electromagnetic forces. The bonds between atoms are made by the electrons and their fields and they are elastic. We just usually don't notice the elasticity usually because we can't handle individual electrons by hand and once we have large quantity of electrons, humans are too weak to notice them easily. However, with things like a thing, long piece of wire (like in a compound bow), you can directly observe and feel it.
When you move a stick around, you are actually doing something akin to pulling a massive array of linked chains around. It does take a small amount of time for the electromagnetic force from the first atom to propagate all the way to the far end of the chain but you don't notice because it's really a short amount of time.
When you push your super magic 2.5ly stick, pretending you are strong enough, it would just buckle near you and that would be the end of the story. The far end of the stick would never move.
However, let's pretend it doesn't buckle because that's more fun.
Even though the stick is fairly light per linear foot, because of it's immense length, you would have to be super strong to push it 5 inches withing a few seconds. Your massive strength would pulverize the stick near you immediately.
But's let's pretend that doesn't happen because the stick is indestructible.
The stick will compress an large amount near you. Withing the first few miles (pure guess) of the stick, there will be a lot of compression. Within the first few feet, you just created an incredible shock wave. What happened is you just created a seismic within the stick. This would eventually spread out into a super low frequency sound wave. Assuming your stick is steel and you took 3 seconds to make your 5" move, because steel has a speed of sound of 20000ft/sec, you would create a single compression wave of around 11 miles long.
This wave would lose energy as it traveled the immense distance so it would never reach the star. However, let's pretend that never happens. The wave would move at the speed of sound and take about 2.95 million years to get to the star.
Now if the stick were perfectly rigid, it would get there immediately. However, since there is a limit on the speed of light, you can deduce that it is impossible for anything to be truly rigid in this universe.
Some people are saying that it would be possible to touch the star because nothing is moving anywhere near the speed of light in this scenario. They are wrong. The only reason an atom in a stick will move is because the atom before it moved first. But the atoms are linked by electromagnetic fields. Atoms connected move by having one atom move, that pushes the electromagnetic field closer to the other atom, and that other atom moves. When you first move a stick, at the very first instant of time, the fist atom moves but every atom beyond the speed of light sphere of that first atom hasn't felt the movement from that first atom yet. The atoms a little ways away from the first atom have no way of knowing that anything has even happened yet.
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