Re-sewing up seams is easy enough - use mattress stitch. Sometimes the trickiest part of mending an old hand-knit is finding new yarn to match the old. Going for a thinner yarn is always a good idea - I used a lace weight variegated yarn (Schoppel-Wolle) selecting the parts where the green was almost the same shade.
Sewing up the under-arms again - and in this photo you can see some old mends done by someone else a long time ago.
I left one of those old mends & strengthened it but re-did the other - and this photo shows the front of the jumper with all mends and strengthening done. This is pre-washing - blocking and some of those specks of paint didn't come out with a simple wash.
Such an old, much worn and loved knit will never look brand new again - but it can be given many more years of usefulness.
Here is how to mend an area that has gone to ladders.
First - catch all the stitches at the top and bottom to stop them laddering further. Putting them onto safety pins works well -
Now re-knit up those ladders - and this is the same technique you use if you've dropped a stitch while hand-knitting.
Take one of those stitches from the bottom of the area onto a crochet hook and thinking of those loose strands as rungs on a ladder, work them up, one rung at a time, taking care to get the 'rungs' in the correct order.
In case you are wondering - that isn't even an official crochet hook. Everyone under the age of 12 seems to be making "Loom Bands" these days and most bags of those little elastic bands come with a hook ...
When re-knitting up a ladder you are working a sort of chain stitch - hook goes through the loop of the stitch, hold in front of the lowest 'rung', catch rung and pull it through loop - and it becomes your new loop, hold it in front of the next rung ... ... repeat till you get to the top. Where you will need a sewing needle and matching yarn to stitch together the stitch / loop from your ladder and the corresponding stitch at the top of the area being mended.
Most of these laddered areas start from a hole and you will probably find some broken yarn somewhere in the middle ... sewing needle & yarn will be needed there.
This photo shows the mend pre-washing
And now, after a wash ... a few more washes and I think the mend might be practically invisible.