My Superman lessons come from the 50's. Maybe I'm having false memory syndrome, but Supergirl, as I recall, was able to rescue Superman from a bad kryptonite trip, as needed, due to her immunity.
No, Brainster is correct on this. Supergirl, being from Krypton, had the same vulnerability to Kryptonite -- red, green, and gold -- as Superman, Krypto, and all other Kryptonians.
There was a try-out run for Supergirl in Superman # 123, where Jimmy Olsen is granted 3 wishes. One wish is to create a Supergirl to be Superman's companion. When Superman winds up in a Kryptonite death trap, she flies in to save him, telling him she's immune because she's magic rather than Kryptonian... but she's lying, and actually has suffered a fatal exposure, so Jimmy wishes her back out of existence.
One story you may be thinking of is "Supergirl's Greatest Victory", from Action Comics 262 (her 11th appearance there), in which she tries to make herself immune to Kryptonite (and appears to succeed -- but not really).
As you can see by the cover, Superman is surprised by this: "
>Gasp<! I'm too weak to even crawl away from this Kryptonite meteor. But Supergirl still has her super-strength. How did she gain immunity?" Strictly a one-issue thing.
There were a number of times when Supergirl acquired immunity, but always briefly and temporarily. An early instance occurred during the long serial in which Supergirl lost her powers, was adopted, regained her powers, and was allowed to introduce herself to the world. (See Action 278-285). Prior to that, Superman had required her to stay in the orphanage and keep her super-identity a secret, both because he felt she needed to learn how to use her powers before she announced her existence and because he felt she'd be more useful as his secret emergency weapon.) Her powers had been stolen by a Kandorian scientist, Lesla-Lar, who secretly replaced Supergirl on earth; they were restored by Mr. Mxyzptlk, who also gave her immunity to Kryptonite (since he felt making her more powerful than Superman would be a great joke). That occurred in late '61 /early '62 and may be what you're remembering.
But the immunity only lasted a couple of months. As soon as Mxy returned to his home dimension, the effects of his magic wear off -- so her magically-given powers, including the K-immunity vanished. (But, during the time she had the magic powers, she was able to throw off the effects of Lesla-Lar's ray which had stolen her powers; so when the magic powers wore off, she had her real powers again.)
(She became immune to Kryptonite again in the early 1970s as part of a continuity-muddled revamp -- but she also became a lot less powerful than Superman then, as her powers came and went unpredictably, so that's probably not what you're thinking about.)
Her Super-cat Streaky was invulnerable to Kryptonite, though. He didn't come from Krypton, so had no natural vulnerability to it. Perhaps that's who you're thinking of?
(He got his powers from exposure to X-Kryptonite -- which was green Kryptonite Supergirl had experimented on in an unsuccessful effort to find a way to overcome her and Superman's vulnerability to it. If you look at the scenes in which she does her experiments, she had to use giant tongs, etc., in order to keep a safe distance from the Kryptonite.
Crap, I wish I still had the comics to prove this, but I sold them on Ebay for 2 million bucks.
You're in luck! You can get all those stories for $100 if you want them in great shape, and under $20 if you just want good reading.
The early Supergirl stories are reprinted in
DC's Supergirl archives (you'll need the first 2 volumes, at $50 each). These are hardcover books with good quality color reprints-- about 200 pages of story reprinted per volume.
A less expensive way to get these stories is DC's Showcase series, which typically reprint about 500 pages of stories per volume; the
Supergirl volume lists for $17 and reprints every Supergirl story from 1958 through 1961.
