In the aftermath of the apocalypse, the Known World ceased to use the Gregorian calendar that used to be common in the Nordic countries and most of the rest of Europe, and replaced it with a new one whose properties are only partly known so far.
Excursion: Purposes and Quality Gauges of Calendars[]
Historically, the main purpose of calendars has been to predict the recurrence of days that have special religious and/or agricultural meaning.
Religious reference points often were derived from special astronomical events, which usually, but not necessarily, are tied to the orbit of the Earth around the Sun - i.e., the year. Agricultural dates of importance, like the onset of the bloom on apple trees, may have some variance dependent on the current climate and weather, but are otherwise even more strictly tied to an annual cycle.
(Calendars that use the cycle of the Moon as their principal long-term period instead ("lunar calendar") do exist, though, the most prominent current-day example being the Islamic calendar.)
The length of the year and that of a day do not have an inherent relation to each other (and, in fact, their ratio is subject to variations and a long-term drift). Thus, the primary task of anyone wanting to design a solar calendar is to measure and duplicate the actual length of the year with enough precision that the calendar and the actual year do not wander with respect to each other, like the previously-used Julian calendar did. The Gregorian and Eastern Orthodox calendars have, on average, 365.2425 days in a year, which is sufficiently precise to stay in sync with the planet's movements for millennia.
In order to make precision measurements of a calendar's accuracy, one needs to pinpoint an annual astronomical event and the day in the calendar's system that it is supposed to occur on. In the case of the Gregorian and Eastern Orthodox calendars, this event is the March equinox (apparent position of the Sun crossing the celestial equator from the Southern to the Northern hemisphere) and it is supposed to fall on the 20th of that Month.
Calendars defining a hierarchical system of additional periods of time (like "months", which are only a loose approximation of the lunar cycle in today's solar calendars) and a naming scheme for individual days are important features and have their own gauges of quality (say, unambiguousness of the naming, or ease of computing the number of days between two given dates), but they are nonetheless secondary to having the calendar have a good match to the actual years passing by.
Characteristics of the Post-Rash Calendar[]
The very first panel of the comic, showing the reference date
The reference point of that calendar ("Year 0, Day 0") is the day Iceland shut down traffic from and to the island so as to keep the Rash from spreading to its population. (It is unknown whether that's the actual reason to start the calendar on that day, or merely coincidental.) That day's date as per the pre-Rash calendars is unknown; the storyline of the comic is supposed to begin, with that day, in nondescript "current times". The author did, however, pinpoint that the post-Rash calendar's turn of the years happens in autumn of the Nordics.
The average length of the calendar's year is unknown. Since it has been in effect for only 90 years so far, it is conceivable that the shift resulting from an imprecise length of the year just hasn't accumulated to a noticeable mismatch yet.
Virtually all dates shown so far have been using numbers instead of names, for years, days, and "months", but the only exact dates in the storyline's present of "year 90" that have been given yet are the one on arc 1 page 65
(originally "...8.2.91", changed to "08.2.90" in preparation of the print run) and the draft of the same page
("08.12.90", later corrected for the shifted turn of the years). This is a strong indicator that not only "months" are still in use, but that a year still has twelve of them as well.
On arc 1 page 79, we see that Tuuri and Lalli received the job offer to go on the expedition "three months ago" and that Onni was still disapproving of their trip "one month" before they were set to leave. Again, this confirms that months, in addition to years and days, are still used as lengths of time measurement.
As a timetable in Björköfjärden confirms, there also still is a notion of a "week", outside the format a full date is given in. However, there is no indication yet that months have names besides numbers, or that the days of a week (or any other multiple) have names, like the pre-Rash "weekdays".
The one exception would be the reference to "the 3rd of September" on arc 1 page 387, but it takes place in Finland and even the pre-Rash Finns weren't using the standard Gregorian month names. This suggests that "September" was put in in translation, to ease readers' understanding. (For the records, today's Finns use Finnish month names with a direct reference to the seasons and the appropriate agricultural tasks; September is called Syyskuu, "autumn month".)
Notion of Time[]
Grandfather clock in Nordic Council Lady's anteroom, moments before she called in her "two o'clock appointment"
The pocket watch from Emil's belt pouch
The clocks visible in the year 90 are analog ones (i.e., they have hands, rather than showing digits) and still show the same twelve-segments clock face we use today. Mentions of "two o'clock" and "in fifteen minutes" also confirm that, unlike the calendar, the notion of the time-of-day has not changed.
The pouch watches that are part of the expedition team's gear seem to have a seconds hand, which suggests that the precision of clocks is roughly on par with run-of-the-mill watches available now. The sounds they make do, however, suggest that if the clockwork is still based on a balance wheel, it must be an extremely slow-running one.