Crucio and Imperio can certainly be cast shortly after each other.
When the Dark Lord is resurrected in the graveyard, he uses Crucio on Harry, then very soon after that uses Imperio.
“A little break,’ said Voldemort, the slit-like nostrils dilating with excitement, ‘a little pause … that hurt, didn’t it, Harry? You don’t want me to do that again, do you?’
Harry didn’t answer. He was going to die like Cedric, those pitiless red eyes were telling him so … he was going to die, and there was nothing he could do about it … but he wasn’t going to play along. He wasn’t going to obey Voldemort … he wasn’t going to beg …
‘I asked you whether you want me to do that again?’ said Voldemort softly. ‘Answer me! Imperio!”
- Harry Potter and the Goblet of Fire, Chapter (Priori Incantatem)
The blissful sensation of Imperio is shown to counteract the aftereffect of Crucio. When the Dark Lord cast it on Harry, he no longer felt the aches Crucio left him with, but once Harry fought it off, he felt them again.
“And these words burst from Harry’s mouth; they echoed through the graveyard, and the dream state was lifted as suddenly as though cold water had been thrown over him – back rushed the aches that the Cruciatus Curse had left all over his body – back rushed the realisation of where he was, and what he was facing …
‘You won’t?’ said Voldemort quietly, and the Death Eaters were not laughing now. ‘You won’t say “no”? Harry, obedience is a virtue I need to teach you before you die … perhaps another little dose of pain?”
- Harry Potter and the Goblet of Fire, Chapter (Priori Incantatem)
It’s clear from this the relief Harry felt was a direct result of the Imperius Curse cast on him, showing it can counteract at least the aftereffect of the Cruciatus Curse.
It’s then likely that if Crucio is cast first, Imperio will stop the pain.
It’s never shown what would happen if the Imperius Curse is cast on someone who’s feeling the full effects of the Cruciatus Curse, but it’s possible, from what we see in the graveyard, that it’ll instantly stop the pain and replace it with the blissful feeling.
However, it’s unclear if an Imperiused wizard could feel Crucio.
We don’t see any situation where someone under the Imperius Curse is subjected to the Cruciatus Curse. It’s unclear whether being under the Imperius Curse would make them incapable of feeling it, the pain would break the Imperius, or they would feel the Crucio but remain under the Imperius Curse. There’s also no comparable situation in the books that could give us information to theorize logically what would be most probable.