Clad is a fossil, an archaic past and past participle form of the verb clothe, now generally replaced by clothed, and having the same sense.
Unlike dress, which may be used both transitively and intransitively, clothe may be used only as transitive verb: it requires a direct object, the person who is clothed.
Consequently, your first example, “an old man clad in old clothing”, is grammatical—here clad is the past participle, employed as an adjective. But your other examples are not grammatical. You must provide a direct object:
He clad himself well/neatly.
He clad himself in old clothing.
Clad as a participle is a strictly literary form. It is almost never heard in conversation, and is old-fashioned even in writing.
Clad as a finite verb is distinctly archaic; it should be used only if you are trying to sound mediaeval.†
I advise you not to use either; use clothed instead, in formal contexts, or dressed in either formal or informal contexts.
Do not, however, use dress up as you do in your examples. This is used in only two senses:
To dress in more formal attire than usual: She always dresses up to go to the theatre.
With as, to disguise oneself in clothing: She dressed up as a Klingon for the costume party.
†Since the matter has been raised in the comments, I may mention that clad, as both participle and finite verb, is much more widely known than it is used because it is a very popular word among historical novelists and writers of fantasy. For instance, the folk hero Robin Hood and his Merry Men are almost inevitably described as “clad in Kendal green”. (Not one reader in twenty knows what Kendal green is, but it doesn’t matter.)