To repent can you pour cold water on yourself in the shower. With a pot or something while having teshuva in mind for the sin? Does it do anything? Like I’ve heard people do it before Yom Kippur. Can I do it now as well?
The laws of Teshuva are well documented in the Rambam, in his 10 chapters of the Laws of Teshuva.
Of critical importance are aspects such as stopping to sin, deciding to never sin again and regretting the sin.
The Tefillah Zakkah recited by many before Yom Kippur states that one should fast and inflict oneself - with each sin having a prescribed number of fast-days and certain inflictions.
וְהִנֵּה יָדַעְנוּ כִּי אֲנַחְנוּ מְחֻיָּבִים לְהִתְעַנּוֹת עַל פִּי תִּקּוּנֵי הַתְּשׁוּבָה עַל כָּל חֵטְא וָחֵטְא, וּלְסַגֵּף אֶת גּוּפֵנוּ בִּתְשׁוּבַת הַמִּשְׁקָל נֶגֶד מַה שֶּׁהִתְעַנַּגְנוּ בַּעֲבֵרוֹת.
I suggest you review the above link which is in both Hebrew and English for the prerequisites of Teshuva, and once you've completed those steps, discuss with a Rabbi how to continue, and if there's a need to do so.
There's a custom to dip in the Mikveh on Erev Yom Kippur to cleanse oneself from those impurities that disappear when one goes to Mikveh. If one cannot get to a Mikveh, the custom is to pour on oneself a certain amount of water.
Yes, the rationale is simple, according to Rambam' Hichot Teshuvah 1:4, suffering (in general) is an integral part of Teshuvah for certain sins.
This suffering can be either voluntary or forced (from above). So assuming you transgressed a sin that requires a measure of suffering, you can surely ease it by causing suffer or inconvenience to yourself.
BUT, this is nothing, comparing to learning Torah. You're better off putting your effort into Torah study instead of playing it a saint.