In the 737NG, it is possible to start the other engine without APU if one engine is already started, by routing air from the started engine to the other one.
Is this possible in the Mil Mi 8 helicopter?
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From that I see in the quite detailed description of the helicopter systems in Russian Wikipedia, looks like both engines have independent electric starters (starters-generators) so could be started independently.
There is no mentioning of APU in the description of all systems. They say power plant consists of the two engines. However as mentioned in comments, the helicopter does have the control panel for starting the "Турбоагрегат". It is generally a name for the device that combines turbine engine and electricity generator. APU is called "Вспомогательная силовая установка (ВСУ)" in Russian.