Better start with the not so good news, if the case lining is wool Baize wool is notorious for absorbing stains that will not come out. That said there are some simple methods of cleaning to improve the look of the internal case lining.
Initial warning dont use water based products because if the case is a vintage one the internal lining will be stuck on with animal based glue usually 'Pearl' type, also wool based cloth is notorious for shrinking if it gets wet.
For an initial all over clean I brush Bicarbonate of Soda into the Baize surface using a tooth brush, dont be mean use a goodly amount. Leave case closed for three days then using a Vacuum cleaner remove the powder.
Another product is the shake and Vacuum carpet cleaning product that also works well BUT!!! it leaves the case smelling like a lady of horizontal entertainments boudoir for months and months.
For dark type oil staining Bread saturated with a solvent (my go to first is cellulose thinners) placed in contact with the stain, as the solvent evaporates it draws the oil and dirt out of the cloth into the bread. With care the bread can be folded over the case partitions, also saturated rag application of Cellulose thinners works well for a quick all over clean. Dry cleaning fluids used on a saturated cloth can be used though I do recommend using outdoors only, I only know one brand name 'Perclone' though there are others, these types of solvents are extremely difficult to purchase here due to the glue sniffing craze, though things may be different on your side of the pond.


The only lessons in my life I truly did learn from where the ones I paid for!