I've looked for any reference as to how they might have done it. There are several possible ways to observe the Sun safely with naked-eye observations, such as the camera obscura method (in a dark room, let in a pinhole of light aimed at the Sun, and it'll project the Sun onto a dark surface), or by using some kind of filter, like smoked glass or even very finely pounded metal. Joseph Needham, the esteemed historian of Chinese science, suggests it is the latter:

As in the case of eclipses, it seems likely that from an early date the sun must have been observed through smoky rock-crystal or semi-transparent jade. Indeed, this technique is specifically referred to by Li Shih-Chen, who says that the 'books on jade' mention certain kinds which were used for looking at the sun. A large piece of this was brought, according to one account, as tribute by an embassy from Fu-Sang about +520... But haze due to dust-storms from the Gobi would also have permitted the observation of sun-spots.

This is from Science and Civilisation in China, volume III, p. 436.

In a footnote, he also says:

There is no reason why sun-spot observations could not have been made very early. Many sunspots are easily visible with the naked eye at sunrise or sunset, or reflected on the surface of still waters. They can be seen well through thin haze. A spot about 30,000 miles, or four earth-diameters, across, is large enough to be seen without telescopic aid, and many spots are much larger, even as large as twenty earth-diameters.

Which is to say: there are some plausible suggestions, but there seems to be scant evidence from the time about how they actually observed what they did. It is worth noting that the context of this work was not astronomy per se, but astrology: they were Sun-worshippers who thought that changes to the appearance of the Sun were omens about the present and the future. So it is not what we would think of as a modern scientific discussion where they would talk about methods; it is more of a recording of certain things observed with an interesting to decoding the omens.