Here is my method for preparing and cleaning vintage table linen:
First I unpick the seams, this makes it easier to flatten after washing.
I then repair any damaged embroidery and darn any small holes in the fabric
Then spot treat any marks starting with good old Vanish, a soak in the powder is usually enough but sometimes I will also give it a rub with Vanish stick if it looks like a tough one.
If this doesn't work or the fabric is delicate and you would rather not use a chemical cleaner then a paste of baking powder and lemon juice is also worth trying.
If all else fails then I revert to my grandmothers tried and tested method, sunlight! I put the item out in the sun making sure the mark is prominent. You can hang them on the washing line but I found it works quicker by lying the item flat, your lawn will do fine! If the weather does not allow for this (!) folding the item so the mark in upper most and putting it on a sunny window sill will also work but may take 2 or 3 days.
Then I end with a gentle machine wash using a good quantity of a basic fabric conditioner to remove the stiffness out of the linen. I have found the cheaper the better!