This is Lost in Space (1998).
The film was based on a 1960s TV series of the same name, which was in turn inspired by the 1812 novel, The Swiss Family Robinson. As you noted in the comments, a reboot of the TV series subsequently aired on Netflix in 2018.
The film focuses on the Robinson family, who undertake a voyage to a nearby star system to begin large-scale emigration from a soon-to-be uninhabitable Earth, but are thrown off course by a saboteur and must try to find their way home.
The main protagonists of the film are John Robinson, his wife, Maureen, their two daughters, Judy and Penny, and their son (the youngest child), Will. There are two other main (human) characters, plus a robot.
In one of the opening scenes, Will hacks into a holo-communication from his school teacher to his mother, manipulating the teacher's holographic form so that her head appears on a few other bodies, including that of a cartoon gorilla (notably, the teacher is played by the actress who played Maureen Robinson in the 1960s series).
A bit later in the film, the Robinson family are sent on a space mission to build a hypergate over another planet, so that the remaining population of Earth can be transported to safety, since the Earth itself is becoming uninhabitable as a result of pollution.
They leave Earth in a spaceship, Jupiter 2, which contains upright cryosleep pods (the pods are visible at around the 0:16 mark in the trailer below, and have straps and eye-coverings as you described). The family members place themselves within the pods shortly after launch, expecting to remain within them for the duration of the ten-year journey, but they're soon woken up from cryosleep, except for the elder daughter, Judy, who winds up stuck in her pod longer than the others, and has to be revived with CPR after she's freed.
The Robinsons subsequently encounter time distortions in space, and on the other side of one of those distortions, they find two abandoned spacecraft. They dock Jupiter 2 with one of the ships, and after boarding it, they're attacked by aggressive, alien spider-like creatures. Will stays on Jupiter 2, but he remotely controls a robot that boards the other ship, and uses it to zap many of the alien bugs, as though he were playing a video game.
After that, they meet an older version of Will from 20 years in the future. He tells them who he is pretty much right away, though, so it's not as if that was kept under wraps for very long.