Can I earn 800 euro per month in part time job ?

I'm afraid not. 800 euro a month would be something you could make working full time. However 800 euro a month is way over what is necessary to sustain yourself here.

If you really need that exact amount and no less for whatever reason, you'd need to be a skilled worker to earn that much part time. However, if you're only looking to live comfortably here: around 30 hour work week in simple jobs is enough (40 hour work week would be full time, and 20 hours/week - part time) to cover living expenses in medium-priced neighbourhoods + some spare money.

Could one live in Riga on 800 euro per month?  Would that include rent and food and transportation?

YES but no wage in LATVIA will cover that 800 a month as it is only doctors wage. All who want that live here but work in Norway or Sweden and come back to family four times a year. Average wage for semi skilled is around 400 euro per month. That is if you can find a job. The population is around Million and half and mainly adult or older. With some six million youth living in USA UK full time .Those who want to remain work in Norway, or have a business. I came over from UK with money to restore old house to English  standards. I do not need to work as now pensioned. Living in rural part and out of town or city has one very big advantage it is cheaper by far than living in Riga as rates are as high as UK there. Before you decide to come here find a job no matter what it is first. Then and only then can you budget and see if you can manage to exist here. Food costs are rising high which is silly as I have explained so now Shops here will suffer as Lithuania has Liddel shops now and just outside of Riga with no border control as it is Baltic nations code we can drive over and do months shopping for less than half the cost at Maxima here or Super Netto. Any way one can save cant be bad. Hope that inside view helps.

Jobs here are not like pay in England or elsewhere. If untrained expect low income of Government figures  around 380 to 420 euro fora month at 50 hours a week. All youth left for better wages and conditions. Here a population of under two million and dropping each year as mass migration takes it down from 6 million 5 years or so ago. I waited until I retired to come and build home here.My partner is Latvian. My thought is to exits in rural bliss 300 per month is about it. If Riga and City life about twice that really. Doctors work for around 890 a month so dont expect a great wage. Many work as self employed and some are million a year guys but few in the pecking order. The average Joe is poor and many tricksters exist. Outside of big supermarkets the army of unemployed pester you for change or want to take your trolley back as 50 cents are in it. They are in rural towns .Riga is more the cash place. if I can say my mate from London uk build a holiday village and rather than give in he does bed and breakfast all year in a holiday village he pay 2000 euro in rates a year for and barely exists. I live in rural settings in a small well appointed house self built with vast gardens of 5 acres and pat 100 euro a year all told. You see we have no bins no roads no services one would expect in civil nations. Roads are sand based and pot holed all over. If rains it is porridge . If it snow ice is the main danger as black ice unseen is dangerous so watch your driving. summers are lovely and hot really hot like India some weeks. Always the same. So either you are fried or frozen in winter. Today 11c below is normal. Snow about 12 inches deep carpet. all over as one sheet. We have had to blockade of heavy snows when no one can really go out from house. rest of time you learn fast to drive on ice or stay in bed. My thoughts honesty I have spent all I had and more on this property and thinking back better in UK . All told this is no longer a cheap nation to live in. All want big money just to survive. Food was really cheap 12 years since but now with Eu money all want big spends. I have seen food bill rise to English price almost .in 6 years from 87 euro or Lats then a month for two adults with everything you could hope for. Now 230 euro just about covers that bill