是位 is short for 是一位.
Both 上了年纪 and 身材瘦小 are modifiers (定语) of 老人, but the syntactic structures are different.
Let's start from 身材瘦小 because it is simpler. This is a 主谓短语, that is, a phrase composed of a 主语 (here 身材) and a 谓语 (here 瘦小).
When 主谓短语 is used as a modifier to another noun, we can represent the whole phrase as AB的X where A is 主语, B is 谓语, and X is the noun being modified. The meaning of this phrase is "X whose A is B". Hence 身材瘦小的老人 = 老人 whose 身材 is 瘦小, an old man whose stature is thin and short, as you have already found out. Similar phrases are 经济繁荣的国家 (a country whose economy is prosperous, 包装精美的礼盒 (a gift box whose packaging is exquisite).
上了年纪 a bit more complex. Syntactically it is a 述宾短语. That is, it consists of a 述语 (here 上) and an object (here 年纪).
The most typical examples of 述宾短语 consist of a verb and its object. For example, "参加过二战" with 参加 (participate) as verb, 二战 (WWII) as the object, and 过 is the aspect particle. Hence 参加过二战的老人 = an old man who participated in WWII in the past.
The phrase 上了年纪 also belongs to this structure. The verb is 上 and the object is 年纪. However it is not very typical because the meaning of 上 used in this phrase is only used in a few largely fixed phrases. It means a certain quantity (the object) is comparatively large, or has reached a certain number. Hence 上(了)年纪 = 年纪 is comparatively large, hence "old". Other phrases containing 上 in this sense are:
- 上百, 上千, 上万 can be used as numeral determiners, to mean "a quantity that is at least 100, 1000, or 10000". Hence 上百(个)人 = a group of at least 100 people, 上千条鱼 = A school of at least 1000 fish.
- 上档次 = high-end, high-quality, elegant, etc. For example 上档次的签名, personal signature that is elegant.
- 不上几天 means "within a few days". "收音机用不上几天就坏了" = The radio was only used for a few days, and then it broke.