The article that you link to clarifies in the methods section that there are two different extraction protocols employed, one for 16S information (probably DNA extraction) and one for total RNA. In fact, the 16S data comes from a previous publication. I would suggest following the recommendations in these methods sections with regard to kits, sample treatment, etc., if you are intending to use their methods.
Alternatively, you can do total RNA sequencing, which will strongly bias towards rRNA, which makes up 80% of the total RNA pool. For trying to choose between total RNA sequencing vs. the two independent preps, you can look at this tech note and at the methods sections of any papers whose protocols you want to follow. The reason that rRNA depletion is so common is that if you are not interested in rRNA, it is kind of wasteful to lose 80% of your reads to it. Also, informative rRNA regions such as the hypervariable regions contain most of the taxonomic information, and these are specifically targeted by e.g. 16S amplicon primers, such that very little of the rRNA sequence from total RNA sequencing is useful anyways. Moreover, it is probably preferable to use DNA sequencing to determine taxonomy anyways, as it will probably track actual genome abundance a lot better than rRNA, which is after all probably imperfectly correlated with genome quantity.
If you do two separate preparations, I would strongly recommend not doing the two preparations at two different points, but doing them side by side from the same sample subdivided. It is not at all clear that this was done properly in the linked article. Minor differences in sample preparation are notorious for their effects on microbiome taxonomic/functional composition due to e.g. differences in toughness of gram positive vs. gram negative organisms.