It is actually kind of a far stretch to assume that a depiction of maybe a phalanx means the sumerians had a standing army. In fact, a lot of other evidence contradicts it.

In his very detailled dissertation, Ingo Schrakamp shows that in administrative texts from the early dynastic states of Lagaš and Umma, the army was mostly comprised of temple personal that were liable for military service in wartime. These men were comprised of the temple elite, and received both rations and land for their services, which they would cultivate, but also have various other occupations, ranging from craftsmen to shepherds and fishers, only a small percentage, the aga-us, seem to have had more police-like function. All in all those were several hundred men, since the temple was its own agricultural unit. Most of them would be spearmen in battle, but the texts also mention also axes, slings, chariot- and bowmen.

For the later Akkadian Time, Schrakamp assumes a standing army, supported by seasonal conscripts, for which there are several indicators, but no definite proof.

Furthermore, it is also important to point out that wartime in the ancient orient was, through most of the ages, only in war season. From many royal inscriptions and texts as well as countless administrative texts, we can see that rulers of the variuos civilizations in the area would start military campaigns annually, but would be home before winter, therefore making soldiering a seasonal occupation.

So, in short- no, the sumerian kingdoms probably did not have a standing army, there is not even proof one existed in akkadian times. If the sargonian archives are found - given there is such evidence - that would be the oldest written proof of such an institution.

Sources:

Schrakamp, Ingo, Krieger und Waffen im frühen Mesopotamien. Organisation und Bewaffnung des Militärs in frühdynastischer und sargonischer Zeit, 2010 (Marburg)

Neumann, Hans (Hrsg.), Krieg und Frieden im Alten Vorderasien : 52e Rencontre Assyriologique Internationale, International Congress of Assyriology and Near Eastern Archaeology, Münster, 17. - 21. Juli 2006, 2006 (Münster)

Edit: Wording