Crpc 128

If 04.01.2015 is Sunday, the case will be listed next working day, i.e.,05.01.2015.  But check up the date, because, courts generally do not do such mistakes.  Remaining part of your question, the answer is yes, the court will generally accept your request and take the cheques with a condition if they are dishonoured you may be dealt with severely.  Or otherwise, you make payments along with arrears till that date and move application to remit the future instalments directly in her bank account on the last working day of the month (if that suits you).

Ask your lawyer to check about the exact date.  I am sure the clerk has done some goof up.

And it will be a working day, 

Go on that date and pay some cash.  Never give cheque.  Give cheque only if you are sure if it wont get bounced.  If it get bounced, again she put one more case on you i.e. cheque bounce case, again you will have to roam to court.

Online transfer is best.  Further, section 128 is not meant for the provision of enforcement of maintenance beyond 2 years.  Arrears can be claimed for a period of 1 year only.  Not above that:

A copy of the order of 1 [maintenance or interim maintenance and expenses of proceeding, as the case may be,] shall be given without payment to the person in whose favour it is made, or to his guardian, if any, or to his guardian, if any, or to the person to 2 [whom the allowance for the maintenance or the allowance for the interim maintenance and expenses of proceeding, as the case may be,] is to be paid; and such order may be enforced by any Magistrate in any place where the person against whom it is made may be, on such Magistrate being satisfied as to the identity of the parties and the non-payment of the 3 [allowance, or as the case may be, expenses, due].

The application u/S 125 CRPC is a civil matter clothed under the provisions of criminal law. In execution of order passed by the Magistrate, the non-applicant/ respondent can be sent to civil imprisonment [JAIL] for persistent default in making payment to the applicant/petitioner, that too not more than 30 days at a time and shall be released immediately on making payment of outstanding amount. 

There would not be any conviction and record of sending the non-applicant/respondent to jail in execution of order passed.

Check adjoining judgement also.  All the best.

The date mentioned in the summon is 04.01.2015, which is a sunday.

please let me know what am I supposed to do?

sit idle and let the court find its fault (what the court will do in such a situation)? whether the court will re-issue summon with new date?

thanks