It's hard! I tried to remind myself a lot "babies survive their older siblings" (if there's any case where this isn't true, please don't link it, I would prefer to stay ignorant) and that helped me relax and not worry so much when my 3yo was being overenthusiastic or even rough with the baby sometimes.
We talked about things he could do e.g. stroke/tickle baby's feet, shake a rattle, show the baby a book or a toy, do a funny dance for the baby. It helps once the baby wakes up a bit because he did genuinely find his older brother fascinating.
When he did things like hit him or even bite him, I would try and be curious and say hmm I think you're experimenting, but I can't let you bite/hit. You tell me if you want to do that and I'll find something else to bite. He always did it so gently I think it was purely what will mum do - and I reminded him a lot and basically didn't let him put his mouth anywhere near the baby if I was worried. When he did have a lapse in judgement and hurt the baby I didn't get angry or punish him, I saw it as my issue in terms of a lack of close supervision. (I did point out he was hurt and that I didn't approve, but treated it mildly.) It soon stopped and they have a great relationship now. (They are 4.5 and 1.5 and do still hurt each other out of frustration, like all siblings, but the pure random spite/experimentation stopped).
Also think about activities that you can do one handed and see if you can set up a nice area to do that - do you have a sofa the three of you can snuggle up on to read or watch a film (or endless blippi - whatever), I found simple board games to be great. You don't need to worry about a newborn spoiling a puzzle/game/activity.
YY to sling too, I used this loads until it got really hot again (summer bday so around 9/10 months)