No. Superman is no more vulnerable to magic than any other person in the DC Universe. But given his range of powers and normal ranges of invulnerability, this is counted as a great weakness relatively speaking. While magic-users do not abound in the DC Universe, there are a number of magical phenomenon which can make confrontations against those forces a surprise for the Man of Steel.
Superman is indeed vulnerable to only a tiny slice of phenomenon in the DC Universe, and each iteration was slightly different in their representation of those weaknesses. We will use the Golden Age Superman, Pre-Crisis Superman and Post Crisis Superman as the definitive canon models as reference. DCnU Superman's limits are still being defined, but it is likely they will not deviate too far from the norm.
All canon Supermen:
- Derived their powers from the yellow or higher energy wavelength stars. Orange or red stars do not completely trigger the genetic capabilities that give him his powers. Under orange stars, his powers are half of their strength and under red stars he has no access to his superhuman abilities.
- While this has never been effectively explained, Kryptonite appears to possess the ability to prevent him from accessing the solar energy somehow stored within his cells. In addition to denying him access to his solar powers, prolonged exposure is both dangerous and ultimately toxic to him. Upon the removal of Kryptonite, depending on the exposure, his access to his powers will return relatively shortly. Only a near-death encounter will leave him drained of power.
- possess a weakness to specific wavelengths of radiation. Advanced species who knew of Kryptonians (such as the denizens of the Fourth World, i.e. Darkseid, Highfather and other New Gods, as well as extraterrestrials, particularly during the period of Pre-Crisis Superman who spent a considerable amount of his time in space) used weapons capable of producing radion energy. Such weapons were rare but were capable of affecting Superman.
- Pre-Crisis Superman was also vulnerable to exotic radiations such as Q-radiation and weapons from advanced civilizations that either harnessed red sun wavelengths, or with sufficient energy could overcome his invulnerability making him vulnerable to their effects. Green Lanterns (and presumably the Guardians) can also use their rings to emulate the wavelengths of Kryptonite radiation.
- have less of a weakness and simply an inability to effectively counter magical energies. The true magic of the DCU has always been considered an effect of Chaos and derived from the Lords of Chaos. Such effects of Chaos render a being whose powers are derived from Order less than effective. Superman is not an agent of Order, only that his powers are created from a technological process that is Ordered.
- have a vulnerability to magic that is no greater than any ordinary human's and Superman has been able to fight magical foes, granted with more difficulty, but his overall effectiveness is on par with any of his other villains. One supposition used during the Byrne era's Superman was that magic bypasses his ability to defend against it due to his inability to understand it. Other suppositions assumed his alien heritage left him vulnerable but who can be sure? As usual, DC has never effectively explained why magic works against him when so many other things do not.
Inconsistent storytelling with poorly defined mechanics
Given that magic is not explained effectively, it can be inferred that metahumans such as Black Adam and Captain Marvel who derive their powers from a magical source are more effective against Superman and that would be a misnomer. The strength of Hercules used by Captain Marvel is super-strength on par with Superman's.
His ability to affect Superman is because his strength is as great as Kal-el's not because it is an active magical effect. Unless the Captain or Black Adam are using a directed energy attack such as the Summoning Lightning Discharge against Superman, or a directed magical energy attack, it is a struggle of physical equals. However, the summoning lightning is a powerful directed magical attack and has been used against Superman to stun and weaken him.
Magical objects, weapons and the like are able to affect him and bypass his natural defenses. Hence Wonder Woman's weapons, forged by powerful magical divine beings, are more than capable of harming Superman. He has also been scratched by werewolves and vampires whose magical nature is inherent and required for their very existence.
In most storylines, Superman has been considered to be equal to or slightly ahead of Captain Marvel in terms of raw power and durability. Superman has always been considered to be the DCU flagship hero on Earth in terms of physical superiority with Captain Marvel being considered second to him and titled "the World's Mightiest Mortal". Given that DC prefers to have the two characters locked in a perpetual dead heat because it promotes fanboy enthusiasm, it will likely never be revealed which of them is more powerful.
A note: All of the references used as examples in other answers are non-canon references, Superman/Shazam!: The Return of Black Adam, Smallville and Kingdom Come are all divergent timelines/continuities and do not reflect canon Superman. Nor should they be used as definitive references to the character.