Can I sue my cheating husband in UAE from Philippines?

Extramarital affair is a punishable offence in the UAE

My husband, a Filipino is settled in Dubai, while I am based in the Philippines. I suspect that he has an illicit relationship with another woman, who is also from my country, and is living in with her in Dubai. Is it possible for me to report this matter to the Dubai authorities? How can I file a case against them? Can I sue both from here?
Pursuant to your question (As per my understanding), it may be noted that having an extramarital affair or engaging in a relationship (sexual) without being married is punishable by law, in accordance with Article 356 of the Federal Law No 3 of 1987 on the Issuance of the Penal Law. An unofficial translation of the same reads as follows.

"Anyone indulging in the crime of indecent assault with mutual consent shall be punished by detention for at least one year. However, if the crime is committed against a male or female who is under fourteen years of age, or if the crime is committed by coercion, it shall be punished by temporary imprisonment."
In view of the foregoing statement, if your suspicion has any merit then you may consider filing a complaint against your husband and the woman.
In this regard, it may be further advised that you may seek the assistance of the Al Ameen service of the Dubai Police. Alternatively, you may personally or through an attorney file a complaint against your husband.
The authorities may require you to be present to record your complaint.
Employer not letting off easily
I have been working with a company under an unlimited contract period. My labour card and labour contract expired on June 13 and June 30 respectively. I got a new job offer and wanted to accept the offer. My new employer is willing to double my salary. However, my present employer is making it difficult for me to shift the job. Can I leave my present job without serving my one-month notice period? What are the fines for overstaying in the UAE? What should I do in this case?
It is presumed that your employment is subject to provisions of Federal Law No 8 of 1980 on the Regulation of Labour Relations (UAE Labour Law). Pursuant to your question, it is not very clear as to whether you have worked under a limited period contract or an unlimited contract.
You mentioned that your employment contract was of unlimited duration and then you said your labour contract expired on June 13, 2015. It may be noted that if there is an end date mentioned on your employment contract, then your employment is of limited duration.
And, pursuant to the same, it may be noted that since you were actually working under a limited employment contract, you may not be required to serve a notice period with your present employer, but you may have to compensate the employer for an amount of up to your 45 days' salary.
This is in accordance with Article 116 of the Labour Law, which states: "Where a contract is revoked by the worker for reasons other than those specified in article (121), he shall be required to compensate the employer for any prejudice the latter sustains as a result: provided that the amount of compensation shall not exceed half the worker's remuneration for three months or the residual period of the contract, whichever is shorter, unless the contract contains a provision to the contrary."
However, if your employment contract is of unlimited period, you may be required to serve a notice period for a minimum of 30 days. This is in accordance with the provisions of Article 117(1) of the Labour Law which states: " Both the employer and the worker may terminate a contract of employment of unlimited duration for a valid reason at any time following its conclusion by giving the other party notice in writing at least 30 days before the termination."
Further, it may be noted that if the said notice period gets reduced (either by the employer or the employee), then the party concerned must compensate the other party for the reduced number of days. This is in accordance with Article 119 of the Labour Law that states:
"Where an employer or a worker fails to serve or reduces the notice period of the termination of contract, the party obliged to give notice will have to pay the other party compensation, called "compensation in lieu of notice".
The said compensation shall be equal to the worker's remuneration with respect to the entire period of notice or the time by which it was reduced. This compensation shall be calculated on the basis of the remuneration last received...on the basis of the average daily remuneration referred to in article 57 of this law."
However, as your labour card and visa have already expired, your present employer cannot force you to continue to work or renew the visa. Therefore, the question of notice period may not arise.
But, irrespective of whether you wish to leave this employment or you wish to take up a new one, you will need to have your present visa formally cancelled and only then can your new employer apply for a visa for you. You may request your present employer to do the needful in this regard.
Also, it is noted that your employment visa expired on June 30. After the expiry of the visa, the employee gets a 30-days grace period to exit the UAE. Overstay fines are imposed upon after the expiry of this grace period.
However, you may contact the office of the General Directorate of Residency and Foreigners' Affairs of the emirate from where your visa was issued for detailed information in this regard.