I am going to unpack the couple layers to this onion of question you have...
Lysa can't punish him, hurt him, keep him as her prisoner
On the contrary, since the death of her Lord Husband, Jon Arryn, Lys has become the Lady Regent of the Vale of Arryn. Remember this is a feudal society, and as such Lord have a direct rule over those living on their land (think of federal vs. local governments). It is well within her power to punish/imprison those who are on her land, even if they were brought there from the outside.
why does Catelyn go to her sister in the first place?
One, because it was not expected. After Catelyn arrests Tyrion she spreads a false rumor that they would return to Winterfell. She knows the Lannisters will receive this information as there were many patrons at the inn, and once they do they will be headed in the wrong direction.
Two, it is the safer place to be. Lysa is family and was also the one warn her about the "Lannister plot". The Eyrie is a heavily fortified location, particularly with the Gates of the Moon guarding the entire Vale. A Lannister army trying to free Tyrion would have little chance of reaching him.
What does Catelyn expect her to do?
Catelyn expects Lysa to keep her safe until a proper trial can be arranged. Since Ned is Hand of the King, he would need time to gain support from the King and stop a direct Lannister retaliation. A convoy of some sorts would need to be established to either bring the trial to Tyrion or to bring Tyrion back to Kings Landing safely and securely.
Was all of that a show of impulsive behaviour?
Not really. The initial act of taking Tyrion prisoner hostage on the spot was pretty impulsive, but the rest (as described above) seems reasonably well thought out. Tyrion's capture essentially serves as leverage. I don't think Catelyn believes that Tyrion was the one who directly harmed Bran, but since he is a Lannister (and technically the Tywin's heir) Tyrion is good hostage to at least let Catelyn and Lysa bring their concerns about the "Lannister plot" (i.e. murdering Jon Arryn, overtaking the throne) to light.