Is it possible to re enter to UK after get refused for the first time for visa-free passport holder?
In general, when one is refused entry to the UK as a visa-free national, it is recommended that they subsequently obtain a visa prior to attempting a re-entry. While not a requirement, the entry refusal is a "black mark" on ones' record and a visa will "wipe clean" the record.
As mentioned, however, it is not a requirement: if you believe that you have addressed the issues that you were originally refused entry for, you are free to re-attempt a visa-free entry. You will, however, face stronger scrutiny at the border in the face of your previous refusal (especially since it sounds as though it was relatively recent.) Furthermore, if you are refused again, these black marks have a "compounding" effect, if only to cause the appearance of a person who is "desperate" to gain entry.
What should I prepare to go in UK tomorrow?
You should bring with you the same documentation which you would submit for a visa application. This is general advice; most times when entering countries visa-free you won't be asked for it, but especially after a previous refusal you must be prepared to "make your case" as they say with strong evidence to back your claims. Broadly speaking, such documentation would include bank statements (to prove you can support yourself), proof of hotel bookings or other accommodation arrangements, travel or health insurance (as necessary), etc. You may refer to the UK Government's guidance for visa applications for a discrete list of documentation requirements. Visitor: Supporting Documents Guide
In addition, you should also bring documentation that addresses the deficiencies in your initial entry application which caused your earlier refusal. You've indicated that you've addressed these deficiencies (eg. purchased a ticket to leave to Cologne, etc.), so you should bring with you documentation of these remedies.
As mentioned, of course, you will come under increased scrutiny when arriving since you do have a very recent refusal. If you are refused again, it is highly recommended that you apply for a Standard Visitor Visa before attempting again.