Trapped in Space (1968) by Jack Williamson.
Protagonist Jeff Stone travels to the newly discovered star Topaz to find out what happened to his brother Ben. He discovered there were no planets around Topaz, just a huge asteroid belt. The native aliens were large spider-like creatures. Jeff eventually finds his brother and the rest of his crew. They have all taken deep-sleep shots, a drug that puts the users into suspended animation to conserve oxygen. Ben's deep sleep hypo broke off the needle in his arm.
The book can be borrowed from the Internet Archive here. It was published in 1968, within your date range.
There are actually two species of spider-like aliens involved. Early in the book on page 21, we meet "Buzz Dozen-Dozen". He is a friendly telepathic alien about the size of a dog, but with many legs and described as looking "a little like a blown-up spider". The book has pencil illustrations and there is a picture of Buzz on page 23. He is the adopted "brother-sister" of Lupe Flor, a fellow member of the rescue crew.
The second species of spider-like aliens are the much bigger vacuum-dwelling "Rock Hoppers". They are described on page 114 as having five arms that
...were bright as silver, but the body was black - black, round, flat and with no head.
The Rock Hoppers weave vast webs in space, one of which captured Ben's ship. Jeff ends up fighting one of them in space, but when it is at his mercy he finds he cannot bring himself to kill it. In turn it rescues him. With the help of Buzz, who has similarities to both the Hoppers and the humans, peace is made.
On page 138 Jeff finds his brother Ben. He and the rest of his crew had put themselves into deep sleep in the hope of later rescue.
He found the broken needle stuck through the sleeve of the jet suit into Ben's frozen arm.
The fact that the needle broke off in Ben's case puts it in doubt that he will recover, but he does.