with regard to the running off, can you emphasise about how this is not how 'big boys' behave?
My two children are teenagers now, but my eldest was 4 and was one term away from starting school when my youngest was born, and I also work with children so have dealt with lots of different types of behaviour over the years.
I have found that children who are in the last stages of pre-school often are desperate to be like other 'big girls and boys' that they see. If it's the case, then this can be used to help weed out certain behaviours (taking into account reasonable expectation of them, since they are only four after all!). For example, when my eldest used to run off if we went out I would tell her that 'only big girls get to walk by themselves' so if she wasn't going to behave like a 'big girl' (i.e. by running off) then she would have to walk and hold my hand or the pram. If the behaviour deteriorated further into a tantrum, I would tell her that she will need reins / wrist strap like a baby, because she is behaving like one. (and only on a couple of occasions did this continue to the point where we actually got the wrist strap out and used it so that she had to walk with me. Good behaviour constantly reinforced with praise such as "well done, you're walking nicely by yourself, just like that big girl over there" (whilst pointing out an 8 year old, or something!). Just to make it clear, it's not about using threats, such as 'stop that or I'll use the reins" more about laying out your expectations (and consequences if he doesn't behave) before you set off.
Can you also use a similar tactic for the garden? (only big boys are allowed outside in the garden without their mummies / little boys need mummy outside to watch them / oh dear mummy can't come out right now so you will have to come in if you throw stones as that's what babies do - because they don't know that throwing stones can hurt someone) .
I still use this tactic now (to some degree) in my dance classes as they all want to be like the big girls and boys in the next class up. It's very helpful for teaching the children not to swing on ballet barres or run about when they should be listening, or whining (seems to be a thing for some children in Reception / Year 1 still). I usually ask the child if they have come to the wrong dance class today, because only children in my tots class do that (whatever it is I don't want them to do).
The really really important thing though, is not use a negative tone or make direct remarks like 'you are being a baby, stop that' or 'only babies do that'. Instead, a calm, no nonsense ' you'll have to come inside right now if you don't want to play nicely by yourself like a big boy'. By using the words 'if you don't want to' it makes the child realise that they are making a choice about their behaviour (as opposed to saying 'if you can't play nicely' which implies that it's beyond their control). And if they don't comply, follow through, either bring him inside (without showing anger - I get it, I was often seething with my eldest when she played up) and perhaps tell him 'maybe tomorrow you can go out and play by yourself like a big boy' or go sit outside with him and supervise 'because little boys need mummy to watch and make sure that they don't throw stones and hurt someone'. (again reinforcing to him the reason why we don't throw stones).
Also, show him all the things he can do that his sister can't - such as helping to put cutlery away in the drawer (babies might hurt themselves so sister can't help), do colouring at the table (babies put crayons in their mouths so sister has to play with the baby toys), go to big school (sister is too little, she'd cry for mummy, and would miss out on doing all of the fun things you get to do at big school!). Put a positive spin on everything he can do, that his sister can't yet, and get him to think that you really want to treat him differently to his sister because he is a big boy and capable of doing all these things.
It sounds like that he is bored, ready for school and attention seeking to try and get your undivided attention, which is normal. I'm sure that with lots of praise and positive reinforcement he will outgrow the behaviour once he is settled at school.