In short answer: I afraid that the exact primary sources for OP’s question are not available.
While it would be definitely better to wait for the answer from the specialist, I just make a brief note with some scanned iconographic sources (sorry for very low-resolution images).
Who was garrisoned in fortresses in Norway at that time?
While the deed of Norwegian (actually Vestlanders [Western Norwegians]) local peasant militia under the command of Erik Rosenkrantz (1519-75), Danish governor (lensherr) of Bergenhus is very famous, the Nordic Seven Year’s War (1563-70) exposed the fundamental weakness of Norway at that period, sometimes called as ‘the fifth province of Dano-Norwegian Kingdom’: There was no kingdom-wide defense system in Norway after medieval leding system, conscription of farmers as well ships on district base, had been out of trend in the 15th century, The last time we know leding was called was in 1429 when Bergen was attacked (Jón Viðar Sigurðsson & Riisøy 2011: 221). In theory, the aristocrats in Norway should serve the military service to the king in form of cavalry, but the majority of old Norwegian aristocrat family had died out in Later Middle Ages and replaced by the small number of the new Danish (or German) elites. In other words, the king could not expect their contribution too much.
In course of 16th century, the king tried to re-organize this age-old leding militia conscription, but almost every time he had to be faced with the vehement resistance from Norwegian peasants. It is true that Rosenkrantz succeeded in organizing the militia from the volunteers from Western Norway on his own initiative in 1564 where the inhabitants seriously afraid the disruption of the trade activity due to the invasion of the Swedes, but even he had to resort to the severe punishment, and the representative of the farmers sent the complaint to the king in Copenhagen in 1568 (Bagge & Mykland 1987: 109-113). Even if the leding conscription itself had gone out of trend long ago, the farmers remembered its tradition that the calling for the leding was primarily restricted to the defense and not used for the invasion, and also, especially Eastern Norwegian peasants were very unmotivated to fight with their Eastern neighbors (Rian 2016: 134f., 137). First in the Kalmar War (1611-13) King Christian IV of Denmark-Norway tried to use the Norwegian peasant militia into the field as a part of the Invading army into Sweden, but it also turned out the militia did not function well without effective commanding structure. They lacked not only the military hierarchy, but also the entire military training that became increasingly important in contemporary continental Europe (‘the military revolution’).
Thus, I suppose that the ‘Norwegian soldiers’ in 1580s, the peaceful period between the two wars between Denmark and Sweden over the hegemony in the Baltic, were actually just a small number of Danish garrison troops, not the conscripted Norwegian peasants (some of them were conscripted to the navy in Copenhagen, though, and got good reputation of ‘better sailors than Danish counterparts’). Even in the time of Kalmar War, the total number of soldiers in Norwegian fortresses might be very small, maximum ca.100 (Cf. Bagge & Mikland 1987: 135). It is likely that the garrison troops essentially equip as the similar armory as their Danish colleagues, though those in Norway in 1580s probably enjoyed even fewer resources due to the status of Norway within the Union: the massive re-construction of the fortress of Båhus and Akershus in Oslo [then Christiania] after the Nordic Seven Year’s War had not been conducted until 1590s, and neither indigenous artillery production site nor the arsenal had been built in the 16th century Norway (Cf. Lind 2013).
The Contemporary Images of Danish (= Norwegian?) Soldiers in the Late 16th century
These iconograpic sources suggest that the Danish soldiers in Norwegian fortress equipped some long barrel-firearms (arquebus?) and hallberd, though I'm not sure about the outfit. King Christian IV also specified these two kind of weapons in his Norwegian Law in 1604, basically the Danish translation of the 13th century Norwegian Law of the Realms (1274) with some updates, and the notice of the weapons belonged to the latter, updated parts. While the law mentions the traditional axe as a weapon for the poor cottar that he should bring with when conscripted, these two new kind of weapons, together with bullets, gunpowder, and a sabre, replaced the older armory like a bow, a shield, a sword, and spear (Christian 4.s Norske Lov, II-10f., in Hallager & Brandt 1855: 31-33). While I cannot say the distribution of such weapons in Norway can date further back to 1580s based on this updated clause, King Christian IV at least seemed to assume that they were also available to the Norwegian peasant as well as cottar by the beginning of the 17th century.
That's all I can find for OP's topic for now.
References:
[edited] typo-fixes.