When electron accelerates, there occurs a propagated ripple on it's electric field. But when it moves constantly, does the field "follow it", i.e. changes instantly? How does it deals with the fact that nothing can travel faster than speed of light?
Yes, in a sense, the field "instantly" moves together with it's source (if this source moves uniformly).
There is no aberration of forces. For example, an illuminated charged sphere and you are both approaching each other in uniform inertial motion along paths that do not collide. Light coming from the sphere will appear approaching you at relativistic aberration angle $\sin\alpha = v/c$. However, the electromagnetic force of attraction to the sphere does not experience aberration. It points directly toward the actual position.
So, theoretically you can always know the actual position of the charge and follow it.
It is very simple to understand it by swapping frames. Just think about a charge "at rest" , which disseminates the field and an observer (a test particle), who moves in this field. It makes clear, that the direction of the electrical force exerted by the sphere on the test particle points directly toward the actual position of the sphere.
That does not mean that the force propagates infinitely fast. The force on a test particle at any given instant is due to the electromagnetic field in the immediate vicinity of the particle at that instant.
Suppose that I take a huge piece of paper, many light-years in size, and inscribe on it a set of lines that all intersect at one point. I could make the angular spacing uniform, in which case this could be a picture of the field lines of a charge in its rest frame, but for the purposes of answering your question this doesn't actually matter. The angular spacing can be random.
If the paper is moving toward the right, then an observer far away, looking at the part of the paper near them, can look at the paper and see a line pointing at a certain angle, and then if they check back an hour later, they will see a line pointing at a different angle. There is nothing in this that violates relativity, and no information is being propagated from the center of the paper to the distant parts of it.
What would violate relativity would be if we could grip the center of the paper, change its state of motion, and have the effect be instantly observed far away. That would be analogous to suddenly changing the motion of the charge. If you do that, then the change propagates outward at $c$.
No, the field doesn't change instantly.
While the charge accelerates, the field "ripples". When the acceleration is done, and the charge travels at constant velocity, the field is stable after that.
The ripple moves out at lightspeed, and behind the ripple the stable field moves out at lightspeed too.
The source event which caused the field at some location at some time is indeed in the past of that field event, not at the present location of the charged particle, so this does merit some thought. Here is a quote from section 7.3 of Relativity Made Relatively Easy: "... the field seems to 'know' where the moving source is now. ... It is as if the source gives its 'marching orders' to the field in the form 'line yourself up on my future position, assuming that I will continue at constant velocity'." This is what happens, and it is brought about by ordinary light-speed-limited communication from the source event to any particular field event. One could argue from other considerations that it must come out like this, but it is very nice to see it all hanging together when one carries out the calculation of the solution based on Lineart-Wiechart potentials.