Apophis will fly by Earth around GEO altitude in 2029. May NASA consider launching an Orion spacecraft on a (sub-)orbital trajectory whose apogee is very close to Apophis so that astronauts can study it from close? I guess we can't land on Apophis or enter an orbit around it because it would require too much fuel to "catch up" with its speed and then leave its path and return to Earth again, wouldn't it?
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There's no point in a flyby since there would be no time to conduct any meaningful measurements.
According to Wikipedia, 99942 Apophis will be going about 7 km/s relative to Earth at closest approach on 2029-04-13. That means it would also be going about 7 km/s relative to any sub-orbital spacecraft. A spacecraft in a 400 km (LEO) by 31600 km (Apophis altitude) orbit would be going around 2 km/s relative to Earth at apogee (per this orbit calculator), for an asteroid-relative speed of at least 5 km/s. The asteroid is around 400 m in diameter, so if you were watching out the window as it went by, you'd have maybe 2 seconds in which it was visible as anything more than a bright dot.
That's not enough time to even reliably take a photograph that would be superior to what could be done by using a powerful telescope on the ground or in low Earth orbit (not that either of those would be easy, but they'd have a better chance of success). An attempt to physically sample the asteroid would just result in obliteration of the sampling device (if it successfully made contact, rather than missing entirely on the one chance to intercept).
If you want to gather better data than what we can already collect remotely from Earth, you have to match the asteroid's speed and fly with it for an extended period of time (which could include touching down on its surface). But that requires a lot of delta-v (more than landing on the moon), especially if the mission has crew that want to return home.
Update 2023-09-25: Yesterday, a NASA blog post OSIRIS-REx Spacecraft Departs for New Mission announced the intention to send OSIRIS-REx to 99942 Apophis, including rendezvous and orbit, arriving around the time of its close approach with Earth. So now there is a mission to this object, albeit uncrewed!
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1$\begingroup$ By my back of the envelope calculation the entire Apollo/Saturn V stack comes up a bit short on catching Apophis but that's neglecting the Oberth effect from burning almost all of it in low Earth orbit. It would be a one way mission with only a few days of life support even if you could catch it. $\endgroup$ Jul 25 at 3:29
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1$\begingroup$ All one needs is an arbitrarily strong butterfly net and a spacecraft that can handle arbitrarily strong acceleration ;) $\endgroup$– gerritJul 25 at 11:52
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1$\begingroup$ @gerrit Or an arbitrarily strong and stretchy butterfly net... $\endgroup$ Jul 25 at 19:54
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It might be possible in theory, but it would require a dedicated development campaign. It won't happen in practice as the powers that be have no interest in doing it. Most eyes are focused on the Moon (2020's) and the rest are focused on Mars (2030's).
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1$\begingroup$ @Johannes yes there has been some interest, but nothing like that shown for the Moon and Mars, at least recently. IIRC the crewed asteroid mission was a thing under president Obama that was dropped in favour of the Moon and the Venus option was only a study. For the Moon there is currently HLS / NASA funded full scale missions and for Mars there are a range of landers, NASA interest and planning and Elon Musk/SpaceX who are hell bent on it $\endgroup$– SlartyJul 25 at 10:07