If we assume an ideal situation, with a person's identical twin (with the identical egg developed years later), then the early twin's arm being cut off and attached to the younger body, there should be no rejection unless an exotic element is added.
The older body may, for example, attack the younger one due to an autoimmune disease. For example, if a person had the genes predisposing them to Celiac disease, and the older arm had some of the cells producing anti-gluten antibodies, those antibodies could then attack the digestive tract of the person getting the donated limb (effectively causing celiac). An old person with a weakened immune system could have a disease such as CLL, and the young body might successfully recognize the cancerous cells as foreign, so then the young body could potentially destroy any (defective) bone marrow in the arm of the old limb. Or the older may harbor an infection that the younger body recognizes as foreign. But these are exceptions, not the rule. I don't know of a lot of Frankenstein-style experiments like this personally, so it's a bit hard to say definitively how differently aged cloned body parts would react for sure.
Older bodies do tend to have fewer stem cells and shorter telomeres, which may help repair damage. But the role of stem cells and telomeres both in aging is a bit controversial, to sat the least. Almost anything involved in tissue growth and regeneration is also at least in part associated with cancer.
What you are thinking of in terms of cloning is that the differentiated tissues, which have gone through many rounds of replication, are both hard to "un-differentiate" into stem cells that can perform all roles of body cells, AND that "older" cells may have short telomeres which protect against the effects of cellular replication. So the Dolly clone was essentially older, and suffered negative effects of age at a much younger age due to this (this is over-simplifying). If you want to know more, read HERE.