You should always argue for a raise on your own merits instead of focusing too much on what your colleagues are doing.
In most offices, no two employees have truly equal responsibilities and tasks. It's useful and by most accounts beneficial for employees and employers to be transparent about raises and wages, but the fact that a colleague at the same level is earning more does not by itself justify a raise. Perhaps your (future) colleague will cover different responsibilities. Maybe they want to attract people with certain knowledge or experience that you don't have. They could be (thinking of) redefining the role.
When you ask for a raise you should justify it based on the value you've added to the company in the recent past. That can be anything: meeting or exceeding business goals, reducing costs, increasing profits, managing well, obtaining a certification, ...
The salary range they advertised for your colleague can be useful part of the information you should collect to judge your market worth and to adjust your salary expectations but you never want to phrase a request for a raise as "He's going to make more than me, that's so unfair!" Even if it is.
I don't know your situation or the importance of this written warning in your organisation. In some companies such warnings are used as a gentle reminder that you don't have flexible hours and shouldn't go overboard on coming in late. In others this would be the lead-up to a Performance Improvement Plan or dismissal. If it's serious then yes, arguing for a raise is difficult. It's much easier if you can say that you've acknowledged their feedback, took it to heart and haven't been late since.