You won't be able to leave the country with just an electronic passport - airlines specifically require a biometric travel passport. An e-passport is a document for domestic matters, not international flights. So first you'll need to get a travel passport, and then you can plan your trip to Dubai.
As for the visa: Russians actually do get it on arrival in the UAE. It's usually an automatic tourist visa for 30 days that they stamp right in your passport when you enter. You don't need any advance approvals if you're coming in as a tourist. The exceptions are if you're planning to work, study, or you've got some complicated document situation.
My friend had a similar situation - she was redoing her passport in a rush before her trip. She got it done on an expedited basis, then flew to Dubai without any problems, and they stamped her visa right when she arrived. So the main thing is just to get a regular travel passport, and the rest will go smoothly.