It probably goes without saying that there are a lot of details missing from the current picture that we have of Pacific Rim's universe. This is something, in my experience at least, that can be simultaneously liberating and yet incredibly frustrating; in the sense that fan speculation and fanon are relatively unimpeded by the direct contradiction of canon, but at the same time there's an awful lot of blanks left to deal with (some rather more annoying than others). In such situations the best that can be done is drawing on circumstantial evidence to try and at least narrow down the field of possibility, until there's a more direct comment in canon anyway (which may well never come).
If you're still with me, the main purpose of this post is to lay out one such exercise in extrapolation, attempting to get as close to a specific date for the Hardship attack as possible within the limits of what we already know. This, in my opinion, is probably the easiest Kaiju to estimate a Breach date for, having the definitive placement of being only the second Jaeger kill on record.
As with every other Kaiju (excluding, of course, Trespasser, Hundun, Kaiceph and Scissure) Hardship must necessarily have emerged after Karloff, the first to be dispatched via Jaeger. Fortunately that notable date is explicitly specified in canon - 23rd April 2015.
Now, here is where the positioning in the Kaiju line-up is important. The next definitively chronological Kaiju we have is Onibaba, who attacked Tokyo on the 15th May 2016. Thus, in order to preserve the canonical ordering, Hardship has to have arrived prior to this date which allows us to set up a narrower range for the attack - after 23rd April 2015 but before 15th May 2016, broadly leaving 387 days as possibilities.
Another crucial date at this juncture is that of Romeo Blue's launch - 15th December 2015. Given that we know it was the American mark-1 who killed Hardship, the Kaiju in question can't have come through the Breach before that launch. Meaning we can eliminate all but 121 days in the initial range from consideration. (Although one could possibly argue that the kill came fractionally before the official launch, but for the sake of the exercise I'm going to ignore that for the moment).
If we want to narrow that number down further this is where things have to get a little sketchier. Obviously the best person to turn to with regards to Kaiju dating is Dr Gottleib, given the nature of his work as seen in the film. His main findings are that Kaiju attacks, largely, occur at increasingly smaller, and arguably regular, intervals over time. This would certainly seem to imply that the spacing, for the most part, will decrease with each new Kaiju. Now, while his exact statement is that "in the beginning the Kaiju attacks were spaced by twenty-four weeks, then twelve, then six, then every two weeks..." this is somewhat undermined when we look at the given dates of the first five Kaiju;
Trespasser: 10th August 2013
Hundun: 5th February 2014 (179 days later)
Kaiceph: 1st June 2014 (116 days later)
Scissure: 2nd September 2014 (93 days later)
Karloff: 23rd April 2015 (233 days later)
Which is why we couldn't have just counted twenty-four weeks from Karloff's Breach date and called it a night. The only way Gottleib's information really serves this exercise is if we look not backwards, but forwards in chronology. There is exactly five months (or roughly twenty-two weeks) between Romeo Blue's listed launch date and the Onibaba attack which would certainly fit with the assertion that there is a decreasing interval between incidents, at least overall. This early on in the Kaiju War it would seem safe to assume that the interval would be at the larger end of the scale, meaning December.
An exact day is obviously impossible to reliably pin down, the lack of a clear pattern between the earlier Kaiju offers no help there. As such, my best conclusion is that Hardship made land at some point in December 2015, which we could perhaps narrow to the 15th or later if we could safely say that Romeo Blue's official launch preceded its first deployment in combat.