You ask "can" you appeal? The answer is "yes," but because of the time that has passed since your conviction, the appeal would not be an "appeal of right." Rather, you would have to ask the Circuit Court to hear your appeal, as an "appeal by leave." The difference is that if you had filed an appeal within 21 days of your conviction, it would have been an appeal of right and the Circuit Court would have had no choice but to hear and entertain your arguments. Now, the Circuit Court can choose whether or not to hear your appeal. You also have to make sure you comply with and use the correct court rules and comply with the proper procedures, starting with MCR 7.103. You can find the rules online at www.courts.michigan.gov. It would be in your best interest to try to hire an attorney to help you out. Good luck.
The above appeal process posting is not necessarily correct. If you pleaded guilty as opposed to being convicted at trial, you never had an "appeal as of right." A guilty plea forecloses your ability to file an "appeal as of right" and only allows for an "appeal by application." There are time guidelines on this type of appeal as well. So, it depends on how you were convicted; by trial or plea to determine "how" to appeal. It seems that you are going to be well past the filing dealine though, so you'll need to hire an appellate practice attorney to handle your possible filing of a late application.
However, if your defense is that you got a medical marijuana card AFTER you were caught with the dope, it isn't going to matter. Just because you could've gotten a prescription at the time that you were arrested doesn't matter. It's treated the same as with any other medication.