For many completed contests you can go back to problems that were not solved during the contest, or solve already completed problems again. These solutions will not count in the standings and they will not be considered by the jury, but they will be judged on the test server. There is no time limit for post-contest solving.
In order to return to a contest, find it in the list or use a direct link provided by the organizers. If the contest is available for post-contest solving, then the status “Your participation in the contest is completed. You can submit your solutions” will be displayed on the page. The problems are solved the same way as in an ordinary contest.
You can take part in some already-finished contests (if you weren't able to participate in them at the time) via virtual participation mode. This mode is as close as possible to participating in a real contest, in that all rules will apply to you, including the time limit on contest duration.
Virtual participants are not displayed in the official contest standings.